topology - northern illinois universityalinner/math650/topologyiii.pdf · 2016. 5. 4. · this...

Topology

Linner

Topology ininfinitedimensions.

Special topic. Topology

Anders Linneralinner@math.niu.edu

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Topology

Linner

Topology ininfinitedimensions.

Special topic.

Special topic.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let X be a real vector space and denote the zero vector by 0.

There is map (a, x) 7→ ax : R× X → X known as scalarmultiplication.

There is also (x , y) 7→ x + y : X × X → X vector addition.

These maps satisfy the familiar vector space axioms.

A common way to introduce a topology on X is via a norm.

A norm is a map || || : X → [0,∞).

It coexists with scalar multiplication, ||ax || = |a|||x ||, vectoraddition, ||x + y || ≤ ||x ||+ ||y ||, and ||x || = 0 if and only ifx = 0.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let X be a real vector space and denote the zero vector by 0.

There is map (a, x) 7→ ax : R× X → X known as scalarmultiplication.

There is also (x , y) 7→ x + y : X × X → X vector addition.

These maps satisfy the familiar vector space axioms.

A common way to introduce a topology on X is via a norm.

A norm is a map || || : X → [0,∞).

It coexists with scalar multiplication, ||ax || = |a|||x ||, vectoraddition, ||x + y || ≤ ||x ||+ ||y ||, and ||x || = 0 if and only ifx = 0.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let X be a real vector space and denote the zero vector by 0.

There is map (a, x) 7→ ax : R× X → X known as scalarmultiplication.

There is also (x , y) 7→ x + y : X × X → X vector addition.

These maps satisfy the familiar vector space axioms.

A common way to introduce a topology on X is via a norm.

A norm is a map || || : X → [0,∞).

It coexists with scalar multiplication, ||ax || = |a|||x ||, vectoraddition, ||x + y || ≤ ||x ||+ ||y ||, and ||x || = 0 if and only ifx = 0.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let X be a real vector space and denote the zero vector by 0.

There is map (a, x) 7→ ax : R× X → X known as scalarmultiplication.

There is also (x , y) 7→ x + y : X × X → X vector addition.

These maps satisfy the familiar vector space axioms.

A common way to introduce a topology on X is via a norm.

A norm is a map || || : X → [0,∞).

It coexists with scalar multiplication, ||ax || = |a|||x ||, vectoraddition, ||x + y || ≤ ||x ||+ ||y ||, and ||x || = 0 if and only ifx = 0.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let X be a real vector space and denote the zero vector by 0.

There is map (a, x) 7→ ax : R× X → X known as scalarmultiplication.

There is also (x , y) 7→ x + y : X × X → X vector addition.

These maps satisfy the familiar vector space axioms.

A common way to introduce a topology on X is via a norm.

A norm is a map || || : X → [0,∞).

It coexists with scalar multiplication, ||ax || = |a|||x ||, vectoraddition, ||x + y || ≤ ||x ||+ ||y ||, and ||x || = 0 if and only ifx = 0.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let X be a real vector space and denote the zero vector by 0.

There is map (a, x) 7→ ax : R× X → X known as scalarmultiplication.

There is also (x , y) 7→ x + y : X × X → X vector addition.

These maps satisfy the familiar vector space axioms.

A common way to introduce a topology on X is via a norm.

A norm is a map || || : X → [0,∞).

It coexists with scalar multiplication, ||ax || = |a|||x ||, vectoraddition, ||x + y || ≤ ||x ||+ ||y ||, and ||x || = 0 if and only ifx = 0.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let X be a real vector space and denote the zero vector by 0.

There is map (a, x) 7→ ax : R× X → X known as scalarmultiplication.

There is also (x , y) 7→ x + y : X × X → X vector addition.

These maps satisfy the familiar vector space axioms.

A common way to introduce a topology on X is via a norm.

A norm is a map || || : X → [0,∞).

It coexists with scalar multiplication, ||ax || = |a|||x ||, vectoraddition, ||x + y || ≤ ||x ||+ ||y ||, and ||x || = 0 if and only ifx = 0.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The metric associated with a norm is given byd(x , y) = ||y − x ||.

This metric topology is referred to as the norm-topology.

If each Cauchy sequence converges, then the space is said to becomplete and X is said to be a Banach space.

Assuming the axiom of choice, it is possible to show theexistence of non-continuous linear maps between Banachspaces.

With this in mind, let X ∗ be the collection of all continuouslinear maps from X to R.

The members of X ∗ are known as continuous linear functionals.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The metric associated with a norm is given byd(x , y) = ||y − x ||.

This metric topology is referred to as the norm-topology.

If each Cauchy sequence converges, then the space is said to becomplete and X is said to be a Banach space.

Assuming the axiom of choice, it is possible to show theexistence of non-continuous linear maps between Banachspaces.

With this in mind, let X ∗ be the collection of all continuouslinear maps from X to R.

The members of X ∗ are known as continuous linear functionals.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The metric associated with a norm is given byd(x , y) = ||y − x ||.

This metric topology is referred to as the norm-topology.

If each Cauchy sequence converges, then the space is said to becomplete and X is said to be a Banach space.

Assuming the axiom of choice, it is possible to show theexistence of non-continuous linear maps between Banachspaces.

With this in mind, let X ∗ be the collection of all continuouslinear maps from X to R.

The members of X ∗ are known as continuous linear functionals.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The metric associated with a norm is given byd(x , y) = ||y − x ||.

This metric topology is referred to as the norm-topology.

If each Cauchy sequence converges, then the space is said to becomplete and X is said to be a Banach space.

Assuming the axiom of choice, it is possible to show theexistence of non-continuous linear maps between Banachspaces.

With this in mind, let X ∗ be the collection of all continuouslinear maps from X to R.

The members of X ∗ are known as continuous linear functionals.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The metric associated with a norm is given byd(x , y) = ||y − x ||.

This metric topology is referred to as the norm-topology.

If each Cauchy sequence converges, then the space is said to becomplete and X is said to be a Banach space.

Assuming the axiom of choice, it is possible to show theexistence of non-continuous linear maps between Banachspaces.

With this in mind, let X ∗ be the collection of all continuouslinear maps from X to R.

The members of X ∗ are known as continuous linear functionals.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The metric associated with a norm is given byd(x , y) = ||y − x ||.

This metric topology is referred to as the norm-topology.

If each Cauchy sequence converges, then the space is said to becomplete and X is said to be a Banach space.

Assuming the axiom of choice, it is possible to show theexistence of non-continuous linear maps between Banachspaces.

With this in mind, let X ∗ be the collection of all continuouslinear maps from X to R.

The members of X ∗ are known as continuous linear functionals.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let B ⊂ X be given by B = x ∈ X | ||x || ≤ 1.

A major source of frustration is that if X isinfinite-dimensional, then B is not compact.

An instance of this is given by observing that the sequence tn

in C [0, 1], the space of continuous functions on [0, 1] with||x || = max0≤t≤1 |x(t)|, has no convergent subsequence.

The trouble is that there are too many open sets, so there aretoo many open covers and not all of them have finite subcovers.

An attempted remedy is to introduce the weak topologyassociated with the collection X ∗.

The norm-continuous linear functionals will still be continuousin the weak topology, and this new topology may have ‘fewer’open sets.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let B ⊂ X be given by B = x ∈ X | ||x || ≤ 1.

A major source of frustration is that if X isinfinite-dimensional, then B is not compact.

An instance of this is given by observing that the sequence tn

in C [0, 1], the space of continuous functions on [0, 1] with||x || = max0≤t≤1 |x(t)|, has no convergent subsequence.

The trouble is that there are too many open sets, so there aretoo many open covers and not all of them have finite subcovers.

An attempted remedy is to introduce the weak topologyassociated with the collection X ∗.

The norm-continuous linear functionals will still be continuousin the weak topology, and this new topology may have ‘fewer’open sets.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let B ⊂ X be given by B = x ∈ X | ||x || ≤ 1.

A major source of frustration is that if X isinfinite-dimensional, then B is not compact.

An instance of this is given by observing that the sequence tn

in C [0, 1], the space of continuous functions on [0, 1] with||x || = max0≤t≤1 |x(t)|, has no convergent subsequence.

The trouble is that there are too many open sets, so there aretoo many open covers and not all of them have finite subcovers.

An attempted remedy is to introduce the weak topologyassociated with the collection X ∗.

The norm-continuous linear functionals will still be continuousin the weak topology, and this new topology may have ‘fewer’open sets.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let B ⊂ X be given by B = x ∈ X | ||x || ≤ 1.

A major source of frustration is that if X isinfinite-dimensional, then B is not compact.

An instance of this is given by observing that the sequence tn

in C [0, 1], the space of continuous functions on [0, 1] with||x || = max0≤t≤1 |x(t)|, has no convergent subsequence.

The trouble is that there are too many open sets, so there aretoo many open covers and not all of them have finite subcovers.

An attempted remedy is to introduce the weak topologyassociated with the collection X ∗.

The norm-continuous linear functionals will still be continuousin the weak topology, and this new topology may have ‘fewer’open sets.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let B ⊂ X be given by B = x ∈ X | ||x || ≤ 1.

A major source of frustration is that if X isinfinite-dimensional, then B is not compact.

An instance of this is given by observing that the sequence tn

in C [0, 1], the space of continuous functions on [0, 1] with||x || = max0≤t≤1 |x(t)|, has no convergent subsequence.

The trouble is that there are too many open sets, so there aretoo many open covers and not all of them have finite subcovers.

An attempted remedy is to introduce the weak topologyassociated with the collection X ∗.

The norm-continuous linear functionals will still be continuousin the weak topology, and this new topology may have ‘fewer’open sets.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let B ⊂ X be given by B = x ∈ X | ||x || ≤ 1.

A major source of frustration is that if X isinfinite-dimensional, then B is not compact.

An instance of this is given by observing that the sequence tn

in C [0, 1], the space of continuous functions on [0, 1] with||x || = max0≤t≤1 |x(t)|, has no convergent subsequence.

The trouble is that there are too many open sets, so there aretoo many open covers and not all of them have finite subcovers.

An attempted remedy is to introduce the weak topologyassociated with the collection X ∗.

The norm-continuous linear functionals will still be continuousin the weak topology, and this new topology may have ‘fewer’open sets.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Whereas the members of X ∗ continue to be continuous in thenew topology, it will be harder for other functions to staycontinuous.

Suppose that X is in fact an inner product space with〈 , 〉 : X × X → R.

The norm satisfies ||x ||2 = 〈x , x〉.

Write xk → x∞ when ||xk − x∞|| → 0, a statement about realnumbers.

Write xk →w x∞ when 〈xk , y〉 → 〈x∞, y〉 for each y ∈ X .

This provides a more concrete description of the weak topology.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Whereas the members of X ∗ continue to be continuous in thenew topology, it will be harder for other functions to staycontinuous.

Suppose that X is in fact an inner product space with〈 , 〉 : X × X → R.

The norm satisfies ||x ||2 = 〈x , x〉.

Write xk → x∞ when ||xk − x∞|| → 0, a statement about realnumbers.

Write xk →w x∞ when 〈xk , y〉 → 〈x∞, y〉 for each y ∈ X .

This provides a more concrete description of the weak topology.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Whereas the members of X ∗ continue to be continuous in thenew topology, it will be harder for other functions to staycontinuous.

Suppose that X is in fact an inner product space with〈 , 〉 : X × X → R.

The norm satisfies ||x ||2 = 〈x , x〉.

Write xk → x∞ when ||xk − x∞|| → 0, a statement about realnumbers.

Write xk →w x∞ when 〈xk , y〉 → 〈x∞, y〉 for each y ∈ X .

This provides a more concrete description of the weak topology.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Whereas the members of X ∗ continue to be continuous in thenew topology, it will be harder for other functions to staycontinuous.

Suppose that X is in fact an inner product space with〈 , 〉 : X × X → R.

The norm satisfies ||x ||2 = 〈x , x〉.

Write xk → x∞ when ||xk − x∞|| → 0, a statement about realnumbers.

Write xk →w x∞ when 〈xk , y〉 → 〈x∞, y〉 for each y ∈ X .

This provides a more concrete description of the weak topology.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Whereas the members of X ∗ continue to be continuous in thenew topology, it will be harder for other functions to staycontinuous.

Suppose that X is in fact an inner product space with〈 , 〉 : X × X → R.

The norm satisfies ||x ||2 = 〈x , x〉.

Write xk → x∞ when ||xk − x∞|| → 0, a statement about realnumbers.

Write xk →w x∞ when 〈xk , y〉 → 〈x∞, y〉 for each y ∈ X .

This provides a more concrete description of the weak topology.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Whereas the members of X ∗ continue to be continuous in thenew topology, it will be harder for other functions to staycontinuous.

Suppose that X is in fact an inner product space with〈 , 〉 : X × X → R.

The norm satisfies ||x ||2 = 〈x , x〉.

Write xk → x∞ when ||xk − x∞|| → 0, a statement about realnumbers.

Write xk →w x∞ when 〈xk , y〉 → 〈x∞, y〉 for each y ∈ X .

This provides a more concrete description of the weak topology.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Weak convergence may be expressed by 〈xk − x∞, y〉 → 0 foreach y ∈ X .

It follows from Cauchy-Schwarz’s inequality that

xk → x∞ =⇒ xk →w x∞.

Suppose xk∞k=1 is an orthonormal collection of vectors in X ,then

∞∑k=1

|〈x , xk〉|2 ≤ ||x ||2.

It follows from this that orthonormal sequences convergeweakly to zero,

Since ||xk || = 1, it is not true that xk → 0, so weakconvergence does not imply norm-convergence.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Weak convergence may be expressed by 〈xk − x∞, y〉 → 0 foreach y ∈ X .

It follows from Cauchy-Schwarz’s inequality that

xk → x∞ =⇒ xk →w x∞.

Suppose xk∞k=1 is an orthonormal collection of vectors in X ,then

∞∑k=1

|〈x , xk〉|2 ≤ ||x ||2.

It follows from this that orthonormal sequences convergeweakly to zero,

Since ||xk || = 1, it is not true that xk → 0, so weakconvergence does not imply norm-convergence.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Weak convergence may be expressed by 〈xk − x∞, y〉 → 0 foreach y ∈ X .

It follows from Cauchy-Schwarz’s inequality that

xk → x∞ =⇒ xk →w x∞.

Suppose xk∞k=1 is an orthonormal collection of vectors in X ,then

∞∑k=1

|〈x , xk〉|2 ≤ ||x ||2.

It follows from this that orthonormal sequences convergeweakly to zero,

Since ||xk || = 1, it is not true that xk → 0, so weakconvergence does not imply norm-convergence.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Weak convergence may be expressed by 〈xk − x∞, y〉 → 0 foreach y ∈ X .

It follows from Cauchy-Schwarz’s inequality that

xk → x∞ =⇒ xk →w x∞.

Suppose xk∞k=1 is an orthonormal collection of vectors in X ,then

∞∑k=1

|〈x , xk〉|2 ≤ ||x ||2.

It follows from this that orthonormal sequences convergeweakly to zero,

Since ||xk || = 1, it is not true that xk → 0, so weakconvergence does not imply norm-convergence.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Weak convergence may be expressed by 〈xk − x∞, y〉 → 0 foreach y ∈ X .

It follows from Cauchy-Schwarz’s inequality that

xk → x∞ =⇒ xk →w x∞.

Suppose xk∞k=1 is an orthonormal collection of vectors in X ,then

∞∑k=1

|〈x , xk〉|2 ≤ ||x ||2.

It follows from this that orthonormal sequences convergeweakly to zero,

Since ||xk || = 1, it is not true that xk → 0, so weakconvergence does not imply norm-convergence.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Suppose xk →w x∞ and ||xk || → ||x∞||, then xk → x∞.

The following deeper fact transfers knowledge in one topologyto specific information about a different topology.

A weakly convergent sequence is always bounded.

In infinite dimensions there there are some linear spaces thatare not closed.

The linear map (x1, x2, x3, . . . ) 7→ (x1,12x2,

13x3, . . . ) : l2 → l2,

has non-closed image.

There is no expectation that the weak closure is the same asthe norm-closure.

Nonetheless, they are the same for convex sets, so in particularso are affine subsets, which includes the linear subspaces.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Suppose xk →w x∞ and ||xk || → ||x∞||, then xk → x∞.

The following deeper fact transfers knowledge in one topologyto specific information about a different topology.

A weakly convergent sequence is always bounded.

In infinite dimensions there there are some linear spaces thatare not closed.

The linear map (x1, x2, x3, . . . ) 7→ (x1,12x2,

13x3, . . . ) : l2 → l2,

has non-closed image.

There is no expectation that the weak closure is the same asthe norm-closure.

Nonetheless, they are the same for convex sets, so in particularso are affine subsets, which includes the linear subspaces.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Suppose xk →w x∞ and ||xk || → ||x∞||, then xk → x∞.

The following deeper fact transfers knowledge in one topologyto specific information about a different topology.

A weakly convergent sequence is always bounded.

In infinite dimensions there there are some linear spaces thatare not closed.

The linear map (x1, x2, x3, . . . ) 7→ (x1,12x2,

13x3, . . . ) : l2 → l2,

has non-closed image.

There is no expectation that the weak closure is the same asthe norm-closure.

Nonetheless, they are the same for convex sets, so in particularso are affine subsets, which includes the linear subspaces.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Suppose xk →w x∞ and ||xk || → ||x∞||, then xk → x∞.

The following deeper fact transfers knowledge in one topologyto specific information about a different topology.

A weakly convergent sequence is always bounded.

In infinite dimensions there there are some linear spaces thatare not closed.

The linear map (x1, x2, x3, . . . ) 7→ (x1,12x2,

13x3, . . . ) : l2 → l2,

has non-closed image.

There is no expectation that the weak closure is the same asthe norm-closure.

Nonetheless, they are the same for convex sets, so in particularso are affine subsets, which includes the linear subspaces.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Suppose xk →w x∞ and ||xk || → ||x∞||, then xk → x∞.

The following deeper fact transfers knowledge in one topologyto specific information about a different topology.

A weakly convergent sequence is always bounded.

In infinite dimensions there there are some linear spaces thatare not closed.

The linear map (x1, x2, x3, . . . ) 7→ (x1,12x2,

13x3, . . . ) : l2 → l2,

has non-closed image.

There is no expectation that the weak closure is the same asthe norm-closure.

Nonetheless, they are the same for convex sets, so in particularso are affine subsets, which includes the linear subspaces.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Suppose xk →w x∞ and ||xk || → ||x∞||, then xk → x∞.

The following deeper fact transfers knowledge in one topologyto specific information about a different topology.

A weakly convergent sequence is always bounded.

In infinite dimensions there there are some linear spaces thatare not closed.

The linear map (x1, x2, x3, . . . ) 7→ (x1,12x2,

13x3, . . . ) : l2 → l2,

has non-closed image.

There is no expectation that the weak closure is the same asthe norm-closure.

Nonetheless, they are the same for convex sets, so in particularso are affine subsets, which includes the linear subspaces.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Suppose xk →w x∞ and ||xk || → ||x∞||, then xk → x∞.

The following deeper fact transfers knowledge in one topologyto specific information about a different topology.

A weakly convergent sequence is always bounded.

In infinite dimensions there there are some linear spaces thatare not closed.

The linear map (x1, x2, x3, . . . ) 7→ (x1,12x2,

13x3, . . . ) : l2 → l2,

has non-closed image.

There is no expectation that the weak closure is the same asthe norm-closure.

Nonetheless, they are the same for convex sets, so in particularso are affine subsets, which includes the linear subspaces.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Recall that for r > 0 in a metric space (X , d),Bx(r) = y ∈ X | d(y , x) < r.

Define Bx [r ] = y ∈ X | d(y , x) ≤ r.

It is not surprising that Bx [r ] is closed.

To see this, let xk → x∞ with xk ∈ Bx [r ].

Suppose d(x∞, x) > r and let ε = d(x∞, x)− r > 0.

Since d(x∞, x) ≤ d(x∞, xk) + d(xk , x) ≤ d(x∞, xk) + r , itfollows that 0 < ε ≤ d(x∞, xk), which leads to a contradiction.

Now Bx(r) ⊂ Bx [r ], but the discrete topology shows that theinclusion may be strict!

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Recall that for r > 0 in a metric space (X , d),Bx(r) = y ∈ X | d(y , x) < r.

Define Bx [r ] = y ∈ X | d(y , x) ≤ r.

It is not surprising that Bx [r ] is closed.

To see this, let xk → x∞ with xk ∈ Bx [r ].

Suppose d(x∞, x) > r and let ε = d(x∞, x)− r > 0.

Since d(x∞, x) ≤ d(x∞, xk) + d(xk , x) ≤ d(x∞, xk) + r , itfollows that 0 < ε ≤ d(x∞, xk), which leads to a contradiction.

Now Bx(r) ⊂ Bx [r ], but the discrete topology shows that theinclusion may be strict!

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Recall that for r > 0 in a metric space (X , d),Bx(r) = y ∈ X | d(y , x) < r.

Define Bx [r ] = y ∈ X | d(y , x) ≤ r.

It is not surprising that Bx [r ] is closed.

To see this, let xk → x∞ with xk ∈ Bx [r ].

Suppose d(x∞, x) > r and let ε = d(x∞, x)− r > 0.

Since d(x∞, x) ≤ d(x∞, xk) + d(xk , x) ≤ d(x∞, xk) + r , itfollows that 0 < ε ≤ d(x∞, xk), which leads to a contradiction.

Now Bx(r) ⊂ Bx [r ], but the discrete topology shows that theinclusion may be strict!

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Recall that for r > 0 in a metric space (X , d),Bx(r) = y ∈ X | d(y , x) < r.

Define Bx [r ] = y ∈ X | d(y , x) ≤ r.

It is not surprising that Bx [r ] is closed.

To see this, let xk → x∞ with xk ∈ Bx [r ].

Suppose d(x∞, x) > r and let ε = d(x∞, x)− r > 0.

Since d(x∞, x) ≤ d(x∞, xk) + d(xk , x) ≤ d(x∞, xk) + r , itfollows that 0 < ε ≤ d(x∞, xk), which leads to a contradiction.

Now Bx(r) ⊂ Bx [r ], but the discrete topology shows that theinclusion may be strict!

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Recall that for r > 0 in a metric space (X , d),Bx(r) = y ∈ X | d(y , x) < r.

Define Bx [r ] = y ∈ X | d(y , x) ≤ r.

It is not surprising that Bx [r ] is closed.

To see this, let xk → x∞ with xk ∈ Bx [r ].

Suppose d(x∞, x) > r and let ε = d(x∞, x)− r > 0.

Since d(x∞, x) ≤ d(x∞, xk) + d(xk , x) ≤ d(x∞, xk) + r , itfollows that 0 < ε ≤ d(x∞, xk), which leads to a contradiction.

Now Bx(r) ⊂ Bx [r ], but the discrete topology shows that theinclusion may be strict!

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Recall that for r > 0 in a metric space (X , d),Bx(r) = y ∈ X | d(y , x) < r.

Define Bx [r ] = y ∈ X | d(y , x) ≤ r.

It is not surprising that Bx [r ] is closed.

To see this, let xk → x∞ with xk ∈ Bx [r ].

Suppose d(x∞, x) > r and let ε = d(x∞, x)− r > 0.

Since d(x∞, x) ≤ d(x∞, xk) + d(xk , x) ≤ d(x∞, xk) + r , itfollows that 0 < ε ≤ d(x∞, xk), which leads to a contradiction.

Now Bx(r) ⊂ Bx [r ], but the discrete topology shows that theinclusion may be strict!

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Recall that for r > 0 in a metric space (X , d),Bx(r) = y ∈ X | d(y , x) < r.

Define Bx [r ] = y ∈ X | d(y , x) ≤ r.

It is not surprising that Bx [r ] is closed.

To see this, let xk → x∞ with xk ∈ Bx [r ].

Suppose d(x∞, x) > r and let ε = d(x∞, x)− r > 0.

Since d(x∞, x) ≤ d(x∞, xk) + d(xk , x) ≤ d(x∞, xk) + r , itfollows that 0 < ε ≤ d(x∞, xk), which leads to a contradiction.

Now Bx(r) ⊂ Bx [r ], but the discrete topology shows that theinclusion may be strict!

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In normed spaces the set Bx [r ] is convex.

It follows that Bx [r ] is weakly closed.

The weak topology in infinite-dimensional spaces is notmetrizable.

A subset A ⊂ X is weakly bounded if there is some weaklyopen set U with 0 ∈ U and some sU > 0 such that A ⊂ sUwhenever s > sU .

A subset of a normed space is weakly bounded if and only if itis bounded.

It follows that Bx [r ] is both weakly closed and weakly boundedin a normed space.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In normed spaces the set Bx [r ] is convex.

It follows that Bx [r ] is weakly closed.

The weak topology in infinite-dimensional spaces is notmetrizable.

A subset A ⊂ X is weakly bounded if there is some weaklyopen set U with 0 ∈ U and some sU > 0 such that A ⊂ sUwhenever s > sU .

A subset of a normed space is weakly bounded if and only if itis bounded.

It follows that Bx [r ] is both weakly closed and weakly boundedin a normed space.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In normed spaces the set Bx [r ] is convex.

It follows that Bx [r ] is weakly closed.

The weak topology in infinite-dimensional spaces is notmetrizable.

A subset A ⊂ X is weakly bounded if there is some weaklyopen set U with 0 ∈ U and some sU > 0 such that A ⊂ sUwhenever s > sU .

A subset of a normed space is weakly bounded if and only if itis bounded.

It follows that Bx [r ] is both weakly closed and weakly boundedin a normed space.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In normed spaces the set Bx [r ] is convex.

It follows that Bx [r ] is weakly closed.

The weak topology in infinite-dimensional spaces is notmetrizable.

A subset A ⊂ X is weakly bounded if there is some weaklyopen set U with 0 ∈ U and some sU > 0 such that A ⊂ sUwhenever s > sU .

A subset of a normed space is weakly bounded if and only if itis bounded.

It follows that Bx [r ] is both weakly closed and weakly boundedin a normed space.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In normed spaces the set Bx [r ] is convex.

It follows that Bx [r ] is weakly closed.

The weak topology in infinite-dimensional spaces is notmetrizable.

A subset A ⊂ X is weakly bounded if there is some weaklyopen set U with 0 ∈ U and some sU > 0 such that A ⊂ sUwhenever s > sU .

A subset of a normed space is weakly bounded if and only if itis bounded.

It follows that Bx [r ] is both weakly closed and weakly boundedin a normed space.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In normed spaces the set Bx [r ] is convex.

It follows that Bx [r ] is weakly closed.

The weak topology in infinite-dimensional spaces is notmetrizable.

A subset A ⊂ X is weakly bounded if there is some weaklyopen set U with 0 ∈ U and some sU > 0 such that A ⊂ sUwhenever s > sU .

A subset of a normed space is weakly bounded if and only if itis bounded.

It follows that Bx [r ] is both weakly closed and weakly boundedin a normed space.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Weakly compact subsets of normed spaces are bounded.

In normed spaces weakly convergent sequences are bounded.

In infinite dimensions, weakly open non-empty sets areunbounded.

The norm and the weak topology are the same if and only ifthe space is finite-dimensional.

The weak topology is induced by a metric if and only if thespace is finite-dimensional.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Weakly compact subsets of normed spaces are bounded.

In normed spaces weakly convergent sequences are bounded.

In infinite dimensions, weakly open non-empty sets areunbounded.

The norm and the weak topology are the same if and only ifthe space is finite-dimensional.

The weak topology is induced by a metric if and only if thespace is finite-dimensional.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Weakly compact subsets of normed spaces are bounded.

In normed spaces weakly convergent sequences are bounded.

In infinite dimensions, weakly open non-empty sets areunbounded.

The norm and the weak topology are the same if and only ifthe space is finite-dimensional.

The weak topology is induced by a metric if and only if thespace is finite-dimensional.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Weakly compact subsets of normed spaces are bounded.

In normed spaces weakly convergent sequences are bounded.

In infinite dimensions, weakly open non-empty sets areunbounded.

The norm and the weak topology are the same if and only ifthe space is finite-dimensional.

The weak topology is induced by a metric if and only if thespace is finite-dimensional.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Weakly compact subsets of normed spaces are bounded.

In normed spaces weakly convergent sequences are bounded.

In infinite dimensions, weakly open non-empty sets areunbounded.

The norm and the weak topology are the same if and only ifthe space is finite-dimensional.

The weak topology is induced by a metric if and only if thespace is finite-dimensional.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let X be a set, Y a topological space, and φ : X → Y afunction.

The discrete topology on X would make φ continuous.

The intersection of all topologies on X that make φ continuousis a topology and is known as the weak topology induced by φ.

This technique may be extended to a whole collectionφα : X → Yα, where each Yα is a topological space.

It may be that the spaces Yα are the same but the maps φαare different.

The open sets are unions of finite intersections of setsφ−1α (Uα), where Uα is open in Yα.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let X be a set, Y a topological space, and φ : X → Y afunction.

The discrete topology on X would make φ continuous.

The intersection of all topologies on X that make φ continuousis a topology and is known as the weak topology induced by φ.

This technique may be extended to a whole collectionφα : X → Yα, where each Yα is a topological space.

It may be that the spaces Yα are the same but the maps φαare different.

The open sets are unions of finite intersections of setsφ−1α (Uα), where Uα is open in Yα.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let X be a set, Y a topological space, and φ : X → Y afunction.

The discrete topology on X would make φ continuous.

The intersection of all topologies on X that make φ continuousis a topology and is known as the weak topology induced by φ.

This technique may be extended to a whole collectionφα : X → Yα, where each Yα is a topological space.

It may be that the spaces Yα are the same but the maps φαare different.

The open sets are unions of finite intersections of setsφ−1α (Uα), where Uα is open in Yα.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let X be a set, Y a topological space, and φ : X → Y afunction.

The discrete topology on X would make φ continuous.

The intersection of all topologies on X that make φ continuousis a topology and is known as the weak topology induced by φ.

This technique may be extended to a whole collectionφα : X → Yα, where each Yα is a topological space.

It may be that the spaces Yα are the same but the maps φαare different.

The open sets are unions of finite intersections of setsφ−1α (Uα), where Uα is open in Yα.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let X be a set, Y a topological space, and φ : X → Y afunction.

The discrete topology on X would make φ continuous.

The intersection of all topologies on X that make φ continuousis a topology and is known as the weak topology induced by φ.

This technique may be extended to a whole collectionφα : X → Yα, where each Yα is a topological space.

It may be that the spaces Yα are the same but the maps φαare different.

The open sets are unions of finite intersections of setsφ−1α (Uα), where Uα is open in Yα.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Let X be a set, Y a topological space, and φ : X → Y afunction.

The discrete topology on X would make φ continuous.

The intersection of all topologies on X that make φ continuousis a topology and is known as the weak topology induced by φ.

This technique may be extended to a whole collectionφα : X → Yα, where each Yα is a topological space.

It may be that the spaces Yα are the same but the maps φαare different.

The open sets are unions of finite intersections of setsφ−1α (Uα), where Uα is open in Yα.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A net xλ ∈ X converges to x∞ ∈ X if and only if φα(xλ)converges to φα(x∞) in Yα for each α.

Let Z be yet another topological space.

A function f : Z → X is continuous if and only if φα f iscontinuous for each α.

Let A be a set and X all the real-valued functions on A.

For each a ∈ A let Ya = R.

Define φa : X → R by φa(f ) = f (a).

The weak topology on X is the topology of pointwiseconvergence on A.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A net xλ ∈ X converges to x∞ ∈ X if and only if φα(xλ)converges to φα(x∞) in Yα for each α.

Let Z be yet another topological space.

A function f : Z → X is continuous if and only if φα f iscontinuous for each α.

Let A be a set and X all the real-valued functions on A.

For each a ∈ A let Ya = R.

Define φa : X → R by φa(f ) = f (a).

The weak topology on X is the topology of pointwiseconvergence on A.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A net xλ ∈ X converges to x∞ ∈ X if and only if φα(xλ)converges to φα(x∞) in Yα for each α.

Let Z be yet another topological space.

A function f : Z → X is continuous if and only if φα f iscontinuous for each α.

Let A be a set and X all the real-valued functions on A.

For each a ∈ A let Ya = R.

Define φa : X → R by φa(f ) = f (a).

The weak topology on X is the topology of pointwiseconvergence on A.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A net xλ ∈ X converges to x∞ ∈ X if and only if φα(xλ)converges to φα(x∞) in Yα for each α.

Let Z be yet another topological space.

A function f : Z → X is continuous if and only if φα f iscontinuous for each α.

Let A be a set and X all the real-valued functions on A.

For each a ∈ A let Ya = R.

Define φa : X → R by φa(f ) = f (a).

The weak topology on X is the topology of pointwiseconvergence on A.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A net xλ ∈ X converges to x∞ ∈ X if and only if φα(xλ)converges to φα(x∞) in Yα for each α.

Let Z be yet another topological space.

A function f : Z → X is continuous if and only if φα f iscontinuous for each α.

Let A be a set and X all the real-valued functions on A.

For each a ∈ A let Ya = R.

Define φa : X → R by φa(f ) = f (a).

The weak topology on X is the topology of pointwiseconvergence on A.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A net xλ ∈ X converges to x∞ ∈ X if and only if φα(xλ)converges to φα(x∞) in Yα for each α.

Let Z be yet another topological space.

A function f : Z → X is continuous if and only if φα f iscontinuous for each α.

Let A be a set and X all the real-valued functions on A.

For each a ∈ A let Ya = R.

Define φa : X → R by φa(f ) = f (a).

The weak topology on X is the topology of pointwiseconvergence on A.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A net xλ ∈ X converges to x∞ ∈ X if and only if φα(xλ)converges to φα(x∞) in Yα for each α.

Let Z be yet another topological space.

A function f : Z → X is continuous if and only if φα f iscontinuous for each α.

Let A be a set and X all the real-valued functions on A.

For each a ∈ A let Ya = R.

Define φa : X → R by φa(f ) = f (a).

The weak topology on X is the topology of pointwiseconvergence on A.

Topology

Linner

Topology ininfinitedimensions.

Special topic.Given ε > 0 and a ∈ A, letUf ,ε,a = g : A→ R | |g(a)− f (a)| < ε.

This collection is a subbasis for the topology of pointwiseconvergence.

Suppose A = [0, 1] and consider the set containing only thezero function.

Topology

Linner

Topology ininfinitedimensions.

Special topic.Given ε > 0 and a ∈ A, letUf ,ε,a = g : A→ R | |g(a)− f (a)| < ε.

This collection is a subbasis for the topology of pointwiseconvergence.

Suppose A = [0, 1] and consider the set containing only thezero function.

Topology

Linner

Topology ininfinitedimensions.

Special topic.Given ε > 0 and a ∈ A, letUf ,ε,a = g : A→ R | |g(a)− f (a)| < ε.

This collection is a subbasis for the topology of pointwiseconvergence.

Suppose A = [0, 1] and consider the set containing only thezero function.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

If there is a metric, then this set is closed and it is thecountable intersection C of open sets.

These open sets contain the finite intersection of subbasismembers.

The countable union U of the finite sets is countable and thefunction that is 0 on U but 1 otherwise is both in D and not inD since [0, 1] is uncountable.

The topology of pointwise convergence on [0, 1] is not given bya metric.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

If there is a metric, then this set is closed and it is thecountable intersection C of open sets.

These open sets contain the finite intersection of subbasismembers.

The countable union U of the finite sets is countable and thefunction that is 0 on U but 1 otherwise is both in D and not inD since [0, 1] is uncountable.

The topology of pointwise convergence on [0, 1] is not given bya metric.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

If there is a metric, then this set is closed and it is thecountable intersection C of open sets.

These open sets contain the finite intersection of subbasismembers.

The countable union U of the finite sets is countable and thefunction that is 0 on U but 1 otherwise is both in D and not inD since [0, 1] is uncountable.

The topology of pointwise convergence on [0, 1] is not given bya metric.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

If there is a metric, then this set is closed and it is thecountable intersection C of open sets.

These open sets contain the finite intersection of subbasismembers.

The countable union U of the finite sets is countable and thefunction that is 0 on U but 1 otherwise is both in D and not inD since [0, 1] is uncountable.

The topology of pointwise convergence on [0, 1] is not given bya metric.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Suppose X is a vector space with a norm || ||.

The metric topology d(x , y) = ||y − x || will here be referred toas the norm topology.

Let X ∗ denote the collection of all continuous linear mapsX → R.

Using these maps there is now also the weak topology on X .

The weak topology is Hausdorff.

Suppose x 6= y and let r = 12d(y , x).

Observe that Bx(r) is open and convex.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Suppose X is a vector space with a norm || ||.

The metric topology d(x , y) = ||y − x || will here be referred toas the norm topology.

Let X ∗ denote the collection of all continuous linear mapsX → R.

Using these maps there is now also the weak topology on X .

The weak topology is Hausdorff.

Suppose x 6= y and let r = 12d(y , x).

Observe that Bx(r) is open and convex.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Suppose X is a vector space with a norm || ||.

The metric topology d(x , y) = ||y − x || will here be referred toas the norm topology.

Let X ∗ denote the collection of all continuous linear mapsX → R.

Using these maps there is now also the weak topology on X .

The weak topology is Hausdorff.

Suppose x 6= y and let r = 12d(y , x).

Observe that Bx(r) is open and convex.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Suppose X is a vector space with a norm || ||.

The metric topology d(x , y) = ||y − x || will here be referred toas the norm topology.

Let X ∗ denote the collection of all continuous linear mapsX → R.

Using these maps there is now also the weak topology on X .

The weak topology is Hausdorff.

Suppose x 6= y and let r = 12d(y , x).

Observe that Bx(r) is open and convex.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Suppose X is a vector space with a norm || ||.

The metric topology d(x , y) = ||y − x || will here be referred toas the norm topology.

Let X ∗ denote the collection of all continuous linear mapsX → R.

Using these maps there is now also the weak topology on X .

The weak topology is Hausdorff.

Suppose x 6= y and let r = 12d(y , x).

Observe that Bx(r) is open and convex.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Suppose X is a vector space with a norm || ||.

The metric topology d(x , y) = ||y − x || will here be referred toas the norm topology.

Let X ∗ denote the collection of all continuous linear mapsX → R.

Using these maps there is now also the weak topology on X .

The weak topology is Hausdorff.

Suppose x 6= y and let r = 12d(y , x).

Observe that Bx(r) is open and convex.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Suppose X is a vector space with a norm || ||.

The metric topology d(x , y) = ||y − x || will here be referred toas the norm topology.

Let X ∗ denote the collection of all continuous linear mapsX → R.

Using these maps there is now also the weak topology on X .

The weak topology is Hausdorff.

Suppose x 6= y and let r = 12d(y , x).

Observe that Bx(r) is open and convex.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

One aspect of the Hahn-Banach theorem asserts the existenceof a continuous linear map φ : X → R such that φ−1(a)separates Bx(r) and y for some a ∈ R.

It follows that x ∈ φ−1((−∞, a)) and y ∈ φ−1((a,∞)), withboth sets weakly open.

Every weakly closed set is norm closed.

There is an associated norm, the operator norm, defined on X ∗

given by ||φ|| = supB0[1] |φ(x)|.

Regardless of X , X ∗ is a Banach space.

Every sequence that convergence in norm converges weakly.

Look at |φ(xk)− φ(x∞)| = |φ(xk − x∞)| ≤ ||φ||||xk − x∞||.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

One aspect of the Hahn-Banach theorem asserts the existenceof a continuous linear map φ : X → R such that φ−1(a)separates Bx(r) and y for some a ∈ R.

It follows that x ∈ φ−1((−∞, a)) and y ∈ φ−1((a,∞)), withboth sets weakly open.

Every weakly closed set is norm closed.

There is an associated norm, the operator norm, defined on X ∗

given by ||φ|| = supB0[1] |φ(x)|.

Regardless of X , X ∗ is a Banach space.

Every sequence that convergence in norm converges weakly.

Look at |φ(xk)− φ(x∞)| = |φ(xk − x∞)| ≤ ||φ||||xk − x∞||.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

One aspect of the Hahn-Banach theorem asserts the existenceof a continuous linear map φ : X → R such that φ−1(a)separates Bx(r) and y for some a ∈ R.

It follows that x ∈ φ−1((−∞, a)) and y ∈ φ−1((a,∞)), withboth sets weakly open.

Every weakly closed set is norm closed.

There is an associated norm, the operator norm, defined on X ∗

given by ||φ|| = supB0[1] |φ(x)|.

Regardless of X , X ∗ is a Banach space.

Every sequence that convergence in norm converges weakly.

Look at |φ(xk)− φ(x∞)| = |φ(xk − x∞)| ≤ ||φ||||xk − x∞||.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

One aspect of the Hahn-Banach theorem asserts the existenceof a continuous linear map φ : X → R such that φ−1(a)separates Bx(r) and y for some a ∈ R.

It follows that x ∈ φ−1((−∞, a)) and y ∈ φ−1((a,∞)), withboth sets weakly open.

Every weakly closed set is norm closed.

There is an associated norm, the operator norm, defined on X ∗

given by ||φ|| = supB0[1] |φ(x)|.

Regardless of X , X ∗ is a Banach space.

Every sequence that convergence in norm converges weakly.

Look at |φ(xk)− φ(x∞)| = |φ(xk − x∞)| ≤ ||φ||||xk − x∞||.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

One aspect of the Hahn-Banach theorem asserts the existenceof a continuous linear map φ : X → R such that φ−1(a)separates Bx(r) and y for some a ∈ R.

It follows that x ∈ φ−1((−∞, a)) and y ∈ φ−1((a,∞)), withboth sets weakly open.

Every weakly closed set is norm closed.

There is an associated norm, the operator norm, defined on X ∗

given by ||φ|| = supB0[1] |φ(x)|.

Regardless of X , X ∗ is a Banach space.

Every sequence that convergence in norm converges weakly.

Look at |φ(xk)− φ(x∞)| = |φ(xk − x∞)| ≤ ||φ||||xk − x∞||.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

One aspect of the Hahn-Banach theorem asserts the existenceof a continuous linear map φ : X → R such that φ−1(a)separates Bx(r) and y for some a ∈ R.

It follows that x ∈ φ−1((−∞, a)) and y ∈ φ−1((a,∞)), withboth sets weakly open.

Every weakly closed set is norm closed.

There is an associated norm, the operator norm, defined on X ∗

given by ||φ|| = supB0[1] |φ(x)|.

Regardless of X , X ∗ is a Banach space.

Every sequence that convergence in norm converges weakly.

Look at |φ(xk)− φ(x∞)| = |φ(xk − x∞)| ≤ ||φ||||xk − x∞||.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

One aspect of the Hahn-Banach theorem asserts the existenceof a continuous linear map φ : X → R such that φ−1(a)separates Bx(r) and y for some a ∈ R.

It follows that x ∈ φ−1((−∞, a)) and y ∈ φ−1((a,∞)), withboth sets weakly open.

Every weakly closed set is norm closed.

There is an associated norm, the operator norm, defined on X ∗

given by ||φ|| = supB0[1] |φ(x)|.

Regardless of X , X ∗ is a Banach space.

Every sequence that convergence in norm converges weakly.

Look at |φ(xk)− φ(x∞)| = |φ(xk − x∞)| ≤ ||φ||||xk − x∞||.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Only in finite dimensions are norm topologies and weaktopologies the same.

Only in finite dimensions are weak topologies metrizable.

If xλ →w x∞ is a weakly convergent net in a normed space,then ||x∞|| ≤ lim infλ ||xλ||.

A real-valued function on any topological space is said to belower semi-continuous if f (x) ≤ lim infλ |f (xλ)|, whenever xλ isa net converging to x .

Apparently, norms are weakly lower semi-continuous.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Only in finite dimensions are norm topologies and weaktopologies the same.

Only in finite dimensions are weak topologies metrizable.

If xλ →w x∞ is a weakly convergent net in a normed space,then ||x∞|| ≤ lim infλ ||xλ||.

A real-valued function on any topological space is said to belower semi-continuous if f (x) ≤ lim infλ |f (xλ)|, whenever xλ isa net converging to x .

Apparently, norms are weakly lower semi-continuous.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Only in finite dimensions are norm topologies and weaktopologies the same.

Only in finite dimensions are weak topologies metrizable.

If xλ →w x∞ is a weakly convergent net in a normed space,then ||x∞|| ≤ lim infλ ||xλ||.

A real-valued function on any topological space is said to belower semi-continuous if f (x) ≤ lim infλ |f (xλ)|, whenever xλ isa net converging to x .

Apparently, norms are weakly lower semi-continuous.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Only in finite dimensions are norm topologies and weaktopologies the same.

Only in finite dimensions are weak topologies metrizable.

If xλ →w x∞ is a weakly convergent net in a normed space,then ||x∞|| ≤ lim infλ ||xλ||.

A real-valued function on any topological space is said to belower semi-continuous if f (x) ≤ lim infλ |f (xλ)|, whenever xλ isa net converging to x .

Apparently, norms are weakly lower semi-continuous.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Only in finite dimensions are norm topologies and weaktopologies the same.

Only in finite dimensions are weak topologies metrizable.

If xλ →w x∞ is a weakly convergent net in a normed space,then ||x∞|| ≤ lim infλ ||xλ||.

A real-valued function on any topological space is said to belower semi-continuous if f (x) ≤ lim infλ |f (xλ)|, whenever xλ isa net converging to x .

Apparently, norms are weakly lower semi-continuous.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Write co(A) for the intersection of all convex sets that containA.

Write co(A) for the intersection of all closed convex sets thatcontain A.

Write 〈A〉 for the intersection of all linear subspaces thatcontain A.

Write [A] for the intersection of all closed linear subspaces thatcontain A.

In arbitrary topologies on a vector space it may be that co(A)is different from co(A).

In arbitrary topologies on a vector space it may be that [A] isdifferent from 〈A〉.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Write co(A) for the intersection of all convex sets that containA.

Write co(A) for the intersection of all closed convex sets thatcontain A.

Write 〈A〉 for the intersection of all linear subspaces thatcontain A.

Write [A] for the intersection of all closed linear subspaces thatcontain A.

In arbitrary topologies on a vector space it may be that co(A)is different from co(A).

In arbitrary topologies on a vector space it may be that [A] isdifferent from 〈A〉.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Write co(A) for the intersection of all convex sets that containA.

Write co(A) for the intersection of all closed convex sets thatcontain A.

Write 〈A〉 for the intersection of all linear subspaces thatcontain A.

Write [A] for the intersection of all closed linear subspaces thatcontain A.

In arbitrary topologies on a vector space it may be that co(A)is different from co(A).

In arbitrary topologies on a vector space it may be that [A] isdifferent from 〈A〉.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Write co(A) for the intersection of all convex sets that containA.

Write co(A) for the intersection of all closed convex sets thatcontain A.

Write 〈A〉 for the intersection of all linear subspaces thatcontain A.

Write [A] for the intersection of all closed linear subspaces thatcontain A.

In arbitrary topologies on a vector space it may be that co(A)is different from co(A).

In arbitrary topologies on a vector space it may be that [A] isdifferent from 〈A〉.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Write co(A) for the intersection of all convex sets that containA.

Write co(A) for the intersection of all closed convex sets thatcontain A.

Write 〈A〉 for the intersection of all linear subspaces thatcontain A.

Write [A] for the intersection of all closed linear subspaces thatcontain A.

In arbitrary topologies on a vector space it may be that co(A)is different from co(A).

In arbitrary topologies on a vector space it may be that [A] isdifferent from 〈A〉.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Write co(A) for the intersection of all convex sets that containA.

Write co(A) for the intersection of all closed convex sets thatcontain A.

Write 〈A〉 for the intersection of all linear subspaces thatcontain A.

Write [A] for the intersection of all closed linear subspaces thatcontain A.

In arbitrary topologies on a vector space it may be that co(A)is different from co(A).

In arbitrary topologies on a vector space it may be that [A] isdifferent from 〈A〉.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In normed spaces co(A) = co(A) and [A] = 〈A〉 hold.

Moreover, co(A) ⊃ co(A) and [A] ⊃ 〈A〉, with strict inclusionspossible.

The lesson is that it matters when a closure is applied.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In normed spaces co(A) = co(A) and [A] = 〈A〉 hold.

Moreover, co(A) ⊃ co(A) and [A] ⊃ 〈A〉, with strict inclusionspossible.

The lesson is that it matters when a closure is applied.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In normed spaces co(A) = co(A) and [A] = 〈A〉 hold.

Moreover, co(A) ⊃ co(A) and [A] ⊃ 〈A〉, with strict inclusionspossible.

The lesson is that it matters when a closure is applied.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The weak topology of a linear subspace of a normed space isthe same as the subspace topology induced by the weaktopology on the normed space.

Using the operator norm on X ∗, there is yet another space(X ∗)∗, which is typically written X ∗∗.

There is a copy of X inside X ∗∗.

Let Φ : X → X ∗∗ be defined as follows.

First note that Φ(x) must be continuous and linear on X ∗.

Let Φ(x)(φ) = φ(x), and check that on the unit ball in X ∗ themagnitude is in fact ||x ||.

As seen, the map Φ is in fact an isometric isomorphism.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The weak topology of a linear subspace of a normed space isthe same as the subspace topology induced by the weaktopology on the normed space.

Using the operator norm on X ∗, there is yet another space(X ∗)∗, which is typically written X ∗∗.

There is a copy of X inside X ∗∗.

Let Φ : X → X ∗∗ be defined as follows.

First note that Φ(x) must be continuous and linear on X ∗.

Let Φ(x)(φ) = φ(x), and check that on the unit ball in X ∗ themagnitude is in fact ||x ||.

As seen, the map Φ is in fact an isometric isomorphism.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The weak topology of a linear subspace of a normed space isthe same as the subspace topology induced by the weaktopology on the normed space.

Using the operator norm on X ∗, there is yet another space(X ∗)∗, which is typically written X ∗∗.

There is a copy of X inside X ∗∗.

Let Φ : X → X ∗∗ be defined as follows.

First note that Φ(x) must be continuous and linear on X ∗.

Let Φ(x)(φ) = φ(x), and check that on the unit ball in X ∗ themagnitude is in fact ||x ||.

As seen, the map Φ is in fact an isometric isomorphism.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The weak topology of a linear subspace of a normed space isthe same as the subspace topology induced by the weaktopology on the normed space.

Using the operator norm on X ∗, there is yet another space(X ∗)∗, which is typically written X ∗∗.

There is a copy of X inside X ∗∗.

Let Φ : X → X ∗∗ be defined as follows.

First note that Φ(x) must be continuous and linear on X ∗.

Let Φ(x)(φ) = φ(x), and check that on the unit ball in X ∗ themagnitude is in fact ||x ||.

As seen, the map Φ is in fact an isometric isomorphism.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The weak topology of a linear subspace of a normed space isthe same as the subspace topology induced by the weaktopology on the normed space.

Using the operator norm on X ∗, there is yet another space(X ∗)∗, which is typically written X ∗∗.

There is a copy of X inside X ∗∗.

Let Φ : X → X ∗∗ be defined as follows.

First note that Φ(x) must be continuous and linear on X ∗.

Let Φ(x)(φ) = φ(x), and check that on the unit ball in X ∗ themagnitude is in fact ||x ||.

As seen, the map Φ is in fact an isometric isomorphism.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The weak topology of a linear subspace of a normed space isthe same as the subspace topology induced by the weaktopology on the normed space.

Using the operator norm on X ∗, there is yet another space(X ∗)∗, which is typically written X ∗∗.

There is a copy of X inside X ∗∗.

Let Φ : X → X ∗∗ be defined as follows.

First note that Φ(x) must be continuous and linear on X ∗.

Let Φ(x)(φ) = φ(x), and check that on the unit ball in X ∗ themagnitude is in fact ||x ||.

As seen, the map Φ is in fact an isometric isomorphism.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The weak topology of a linear subspace of a normed space isthe same as the subspace topology induced by the weaktopology on the normed space.

Using the operator norm on X ∗, there is yet another space(X ∗)∗, which is typically written X ∗∗.

There is a copy of X inside X ∗∗.

Let Φ : X → X ∗∗ be defined as follows.

First note that Φ(x) must be continuous and linear on X ∗.

Let Φ(x)(φ) = φ(x), and check that on the unit ball in X ∗ themagnitude is in fact ||x ||.

As seen, the map Φ is in fact an isometric isomorphism.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The linear subspace Φ(X ) ⊂ X ∗∗ is closed if and only if X is aBanach space.

The space Φ(X ) is a completion of X .

A space is said to be reflexive if Φ(X ) = X ∗∗.

Every reflexive normed space is a Banach space.

Every normed space that is isomorphic to a reflexive normedspace is itself reflexive.

Every finite-dimensional space is reflexive.

Hilbert spaces are reflexive.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The linear subspace Φ(X ) ⊂ X ∗∗ is closed if and only if X is aBanach space.

The space Φ(X ) is a completion of X .

A space is said to be reflexive if Φ(X ) = X ∗∗.

Every reflexive normed space is a Banach space.

Every normed space that is isomorphic to a reflexive normedspace is itself reflexive.

Every finite-dimensional space is reflexive.

Hilbert spaces are reflexive.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The linear subspace Φ(X ) ⊂ X ∗∗ is closed if and only if X is aBanach space.

The space Φ(X ) is a completion of X .

A space is said to be reflexive if Φ(X ) = X ∗∗.

Every reflexive normed space is a Banach space.

Every normed space that is isomorphic to a reflexive normedspace is itself reflexive.

Every finite-dimensional space is reflexive.

Hilbert spaces are reflexive.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The linear subspace Φ(X ) ⊂ X ∗∗ is closed if and only if X is aBanach space.

The space Φ(X ) is a completion of X .

A space is said to be reflexive if Φ(X ) = X ∗∗.

Every reflexive normed space is a Banach space.

Every normed space that is isomorphic to a reflexive normedspace is itself reflexive.

Every finite-dimensional space is reflexive.

Hilbert spaces are reflexive.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The linear subspace Φ(X ) ⊂ X ∗∗ is closed if and only if X is aBanach space.

The space Φ(X ) is a completion of X .

A space is said to be reflexive if Φ(X ) = X ∗∗.

Every reflexive normed space is a Banach space.

Every normed space that is isomorphic to a reflexive normedspace is itself reflexive.

Every finite-dimensional space is reflexive.

Hilbert spaces are reflexive.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The linear subspace Φ(X ) ⊂ X ∗∗ is closed if and only if X is aBanach space.

The space Φ(X ) is a completion of X .

A space is said to be reflexive if Φ(X ) = X ∗∗.

Every reflexive normed space is a Banach space.

Every normed space that is isomorphic to a reflexive normedspace is itself reflexive.

Every finite-dimensional space is reflexive.

Hilbert spaces are reflexive.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The linear subspace Φ(X ) ⊂ X ∗∗ is closed if and only if X is aBanach space.

The space Φ(X ) is a completion of X .

A space is said to be reflexive if Φ(X ) = X ∗∗.

Every reflexive normed space is a Banach space.

Every normed space that is isomorphic to a reflexive normedspace is itself reflexive.

Every finite-dimensional space is reflexive.

Hilbert spaces are reflexive.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In addition to the operator norm topology, the space X ∗ maybe given the weak topology.

It is also possible to use the family Φ(X ) to produce yetanother weak topology on X ∗.

This topology is known as the weak∗ topology.

The weak∗ and the weak topology on X ∗ are the same if andonly if X is reflexive.

The weak∗ and the operator norm topology on X ∗ are thesame if and only if X is finite-dimensional.

A linear functional on X ∗ is weak∗ continuous if and only ifφ 7→ φ(xφ) for some xφ ∈ X .

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In addition to the operator norm topology, the space X ∗ maybe given the weak topology.

It is also possible to use the family Φ(X ) to produce yetanother weak topology on X ∗.

This topology is known as the weak∗ topology.

The weak∗ and the weak topology on X ∗ are the same if andonly if X is reflexive.

The weak∗ and the operator norm topology on X ∗ are thesame if and only if X is finite-dimensional.

A linear functional on X ∗ is weak∗ continuous if and only ifφ 7→ φ(xφ) for some xφ ∈ X .

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In addition to the operator norm topology, the space X ∗ maybe given the weak topology.

It is also possible to use the family Φ(X ) to produce yetanother weak topology on X ∗.

This topology is known as the weak∗ topology.

The weak∗ and the weak topology on X ∗ are the same if andonly if X is reflexive.

The weak∗ and the operator norm topology on X ∗ are thesame if and only if X is finite-dimensional.

A linear functional on X ∗ is weak∗ continuous if and only ifφ 7→ φ(xφ) for some xφ ∈ X .

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In addition to the operator norm topology, the space X ∗ maybe given the weak topology.

It is also possible to use the family Φ(X ) to produce yetanother weak topology on X ∗.

This topology is known as the weak∗ topology.

The weak∗ and the weak topology on X ∗ are the same if andonly if X is reflexive.

The weak∗ and the operator norm topology on X ∗ are thesame if and only if X is finite-dimensional.

A linear functional on X ∗ is weak∗ continuous if and only ifφ 7→ φ(xφ) for some xφ ∈ X .

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In addition to the operator norm topology, the space X ∗ maybe given the weak topology.

It is also possible to use the family Φ(X ) to produce yetanother weak topology on X ∗.

This topology is known as the weak∗ topology.

The weak∗ and the weak topology on X ∗ are the same if andonly if X is reflexive.

The weak∗ and the operator norm topology on X ∗ are thesame if and only if X is finite-dimensional.

A linear functional on X ∗ is weak∗ continuous if and only ifφ 7→ φ(xφ) for some xφ ∈ X .

Topology

Linner

Topology ininfinitedimensions.

Special topic.

In addition to the operator norm topology, the space X ∗ maybe given the weak topology.

It is also possible to use the family Φ(X ) to produce yetanother weak topology on X ∗.

This topology is known as the weak∗ topology.

The weak∗ and the weak topology on X ∗ are the same if andonly if X is reflexive.

The weak∗ and the operator norm topology on X ∗ are thesame if and only if X is finite-dimensional.

A linear functional on X ∗ is weak∗ continuous if and only ifφ 7→ φ(xφ) for some xφ ∈ X .

Topology

Linner

Topology ininfinitedimensions.

Special topic.

If X is a Banach space, then a subset of X ∗ is bounded if andonly if it is weak∗ bounded.

A collection A ⊂ X ∗ is bounded if and only if φ(x) | φ ∈ A isbounded for each x ∈ X .

If X is a Banach space, then weak∗ compact subsets of X ∗ arebounded.

If X is infinite-dimensional, then weak∗ open subsets of X ∗ areunbounded.

The weak∗ topology of X ∗ is metrizable if and only if X isfinite-dimensional.

The weak∗ topology of X ∗ is complete if and only if X isfinite-dimensional.

Norms are on dual spaces are weak∗ lower semi-continuous.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

If X is a Banach space, then a subset of X ∗ is bounded if andonly if it is weak∗ bounded.

A collection A ⊂ X ∗ is bounded if and only if φ(x) | φ ∈ A isbounded for each x ∈ X .

If X is a Banach space, then weak∗ compact subsets of X ∗ arebounded.

If X is infinite-dimensional, then weak∗ open subsets of X ∗ areunbounded.

The weak∗ topology of X ∗ is metrizable if and only if X isfinite-dimensional.

The weak∗ topology of X ∗ is complete if and only if X isfinite-dimensional.

Norms are on dual spaces are weak∗ lower semi-continuous.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

If X is a Banach space, then a subset of X ∗ is bounded if andonly if it is weak∗ bounded.

A collection A ⊂ X ∗ is bounded if and only if φ(x) | φ ∈ A isbounded for each x ∈ X .

If X is a Banach space, then weak∗ compact subsets of X ∗ arebounded.

If X is infinite-dimensional, then weak∗ open subsets of X ∗ areunbounded.

The weak∗ topology of X ∗ is metrizable if and only if X isfinite-dimensional.

The weak∗ topology of X ∗ is complete if and only if X isfinite-dimensional.

Norms are on dual spaces are weak∗ lower semi-continuous.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

If X is a Banach space, then a subset of X ∗ is bounded if andonly if it is weak∗ bounded.

A collection A ⊂ X ∗ is bounded if and only if φ(x) | φ ∈ A isbounded for each x ∈ X .

If X is a Banach space, then weak∗ compact subsets of X ∗ arebounded.

If X is infinite-dimensional, then weak∗ open subsets of X ∗ areunbounded.

The weak∗ topology of X ∗ is metrizable if and only if X isfinite-dimensional.

The weak∗ topology of X ∗ is complete if and only if X isfinite-dimensional.

Norms are on dual spaces are weak∗ lower semi-continuous.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

If X is a Banach space, then a subset of X ∗ is bounded if andonly if it is weak∗ bounded.

A collection A ⊂ X ∗ is bounded if and only if φ(x) | φ ∈ A isbounded for each x ∈ X .

If X is a Banach space, then weak∗ compact subsets of X ∗ arebounded.

If X is infinite-dimensional, then weak∗ open subsets of X ∗ areunbounded.

The weak∗ topology of X ∗ is metrizable if and only if X isfinite-dimensional.

The weak∗ topology of X ∗ is complete if and only if X isfinite-dimensional.

Norms are on dual spaces are weak∗ lower semi-continuous.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

If X is a Banach space, then a subset of X ∗ is bounded if andonly if it is weak∗ bounded.

A collection A ⊂ X ∗ is bounded if and only if φ(x) | φ ∈ A isbounded for each x ∈ X .

If X is a Banach space, then weak∗ compact subsets of X ∗ arebounded.

If X is infinite-dimensional, then weak∗ open subsets of X ∗ areunbounded.

The weak∗ topology of X ∗ is metrizable if and only if X isfinite-dimensional.

The weak∗ topology of X ∗ is complete if and only if X isfinite-dimensional.

Norms are on dual spaces are weak∗ lower semi-continuous.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

If X is a Banach space, then a subset of X ∗ is bounded if andonly if it is weak∗ bounded.

A collection A ⊂ X ∗ is bounded if and only if φ(x) | φ ∈ A isbounded for each x ∈ X .

If X is a Banach space, then weak∗ compact subsets of X ∗ arebounded.

If X is infinite-dimensional, then weak∗ open subsets of X ∗ areunbounded.

The weak∗ topology of X ∗ is metrizable if and only if X isfinite-dimensional.

The weak∗ topology of X ∗ is complete if and only if X isfinite-dimensional.

Norms are on dual spaces are weak∗ lower semi-continuous.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Some closed convex subsets of X ∗ are not weak∗ closed.

In 1940 Alaoglu pursued ideas outlined in Banach’s book from1932 and proved that in X ∗, B0[1] is weak∗ compact.

Let BX be B0[1] in X , and BX∗∗ be B0[1] in X ∗∗.

NowΦ(BX )

weak∗

= BX∗∗ ,

where the weak∗ topology has been put to work in X ∗∗.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Some closed convex subsets of X ∗ are not weak∗ closed.

In 1940 Alaoglu pursued ideas outlined in Banach’s book from1932 and proved that in X ∗, B0[1] is weak∗ compact.

Let BX be B0[1] in X , and BX∗∗ be B0[1] in X ∗∗.

NowΦ(BX )

weak∗

= BX∗∗ ,

where the weak∗ topology has been put to work in X ∗∗.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Some closed convex subsets of X ∗ are not weak∗ closed.

In 1940 Alaoglu pursued ideas outlined in Banach’s book from1932 and proved that in X ∗, B0[1] is weak∗ compact.

Let BX be B0[1] in X , and BX∗∗ be B0[1] in X ∗∗.

NowΦ(BX )

weak∗

= BX∗∗ ,

where the weak∗ topology has been put to work in X ∗∗.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Some closed convex subsets of X ∗ are not weak∗ closed.

In 1940 Alaoglu pursued ideas outlined in Banach’s book from1932 and proved that in X ∗, B0[1] is weak∗ compact.

Let BX be B0[1] in X , and BX∗∗ be B0[1] in X ∗∗.

NowΦ(BX )

weak∗

= BX∗∗ ,

where the weak∗ topology has been put to work in X ∗∗.

Topology

Linner

Topology ininfinitedimensions.

Special topic.Let A be a subset of a normed space.

The following are equivalent.

(I) A is weakly compact.

(II) Φ(A) is weak∗ compact.

(III) A is bounded and Φ(A) is weak∗ closed.

Topology

Linner

Topology ininfinitedimensions.

Special topic.Let A be a subset of a normed space.

The following are equivalent.

(I) A is weakly compact.

(II) Φ(A) is weak∗ compact.

(III) A is bounded and Φ(A) is weak∗ closed.

Topology

Linner

Topology ininfinitedimensions.

Special topic.Let A be a subset of a normed space.

The following are equivalent.

(I) A is weakly compact.

(II) Φ(A) is weak∗ compact.

(III) A is bounded and Φ(A) is weak∗ closed.

Topology

Linner

Topology ininfinitedimensions.

Special topic.Let A be a subset of a normed space.

The following are equivalent.

(I) A is weakly compact.

(II) Φ(A) is weak∗ compact.

(III) A is bounded and Φ(A) is weak∗ closed.

Topology

Linner

Topology ininfinitedimensions.

Special topic.Let A be a subset of a normed space.

The following are equivalent.

(I) A is weakly compact.

(II) Φ(A) is weak∗ compact.

(III) A is bounded and Φ(A) is weak∗ closed.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A normed space X is reflexive if and only if B0[1] is weaklycompact.

Eberlein-Smulian

Weak compactness is equivalent to sequential weakcompactness.

A normed space is reflexive if and only if each boundedsequence has a weakly convergent subsequence.

Let X be a normed space and consider the subspace topologyon BX∗ induced by the weak∗ topology on X ∗.

This topology is metrizable if and only if X is separable.

This is particularly remarkable in light of the fact that theweak∗ topology on X ∗ is not metrizable when X isinfinite-dimensional.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A normed space X is reflexive if and only if B0[1] is weaklycompact.

Eberlein-Smulian

Weak compactness is equivalent to sequential weakcompactness.

A normed space is reflexive if and only if each boundedsequence has a weakly convergent subsequence.

Let X be a normed space and consider the subspace topologyon BX∗ induced by the weak∗ topology on X ∗.

This topology is metrizable if and only if X is separable.

This is particularly remarkable in light of the fact that theweak∗ topology on X ∗ is not metrizable when X isinfinite-dimensional.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A normed space X is reflexive if and only if B0[1] is weaklycompact.

Eberlein-Smulian

Weak compactness is equivalent to sequential weakcompactness.

A normed space is reflexive if and only if each boundedsequence has a weakly convergent subsequence.

Let X be a normed space and consider the subspace topologyon BX∗ induced by the weak∗ topology on X ∗.

This topology is metrizable if and only if X is separable.

This is particularly remarkable in light of the fact that theweak∗ topology on X ∗ is not metrizable when X isinfinite-dimensional.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A normed space X is reflexive if and only if B0[1] is weaklycompact.

Eberlein-Smulian

Weak compactness is equivalent to sequential weakcompactness.

A normed space is reflexive if and only if each boundedsequence has a weakly convergent subsequence.

Let X be a normed space and consider the subspace topologyon BX∗ induced by the weak∗ topology on X ∗.

This topology is metrizable if and only if X is separable.

This is particularly remarkable in light of the fact that theweak∗ topology on X ∗ is not metrizable when X isinfinite-dimensional.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A normed space X is reflexive if and only if B0[1] is weaklycompact.

Eberlein-Smulian

Weak compactness is equivalent to sequential weakcompactness.

A normed space is reflexive if and only if each boundedsequence has a weakly convergent subsequence.

Let X be a normed space and consider the subspace topologyon BX∗ induced by the weak∗ topology on X ∗.

This topology is metrizable if and only if X is separable.

This is particularly remarkable in light of the fact that theweak∗ topology on X ∗ is not metrizable when X isinfinite-dimensional.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A normed space X is reflexive if and only if B0[1] is weaklycompact.

Eberlein-Smulian

Weak compactness is equivalent to sequential weakcompactness.

A normed space is reflexive if and only if each boundedsequence has a weakly convergent subsequence.

Let X be a normed space and consider the subspace topologyon BX∗ induced by the weak∗ topology on X ∗.

This topology is metrizable if and only if X is separable.

This is particularly remarkable in light of the fact that theweak∗ topology on X ∗ is not metrizable when X isinfinite-dimensional.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A normed space X is reflexive if and only if B0[1] is weaklycompact.

Eberlein-Smulian

Weak compactness is equivalent to sequential weakcompactness.

A normed space is reflexive if and only if each boundedsequence has a weakly convergent subsequence.

Let X be a normed space and consider the subspace topologyon BX∗ induced by the weak∗ topology on X ∗.

This topology is metrizable if and only if X is separable.

This is particularly remarkable in light of the fact that theweak∗ topology on X ∗ is not metrizable when X isinfinite-dimensional.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Mazur

The closed convex hull of a compact subset of a Banach spaceis compact.

Krein-Smulian

The closed convex hull of a weakly compact subset of aBanach space is weakly compact.

The book ‘An introduction to Banach Space Theory’ by RobertE. Megginson, Springer 1998, describes these topologies, and afew others, particularly well.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Mazur

The closed convex hull of a compact subset of a Banach spaceis compact.

Krein-Smulian

The closed convex hull of a weakly compact subset of aBanach space is weakly compact.

The book ‘An introduction to Banach Space Theory’ by RobertE. Megginson, Springer 1998, describes these topologies, and afew others, particularly well.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Mazur

The closed convex hull of a compact subset of a Banach spaceis compact.

Krein-Smulian

The closed convex hull of a weakly compact subset of aBanach space is weakly compact.

The book ‘An introduction to Banach Space Theory’ by RobertE. Megginson, Springer 1998, describes these topologies, and afew others, particularly well.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Mazur

The closed convex hull of a compact subset of a Banach spaceis compact.

Krein-Smulian

The closed convex hull of a weakly compact subset of aBanach space is weakly compact.

The book ‘An introduction to Banach Space Theory’ by RobertE. Megginson, Springer 1998, describes these topologies, and afew others, particularly well.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Mazur

The closed convex hull of a compact subset of a Banach spaceis compact.

Krein-Smulian

The closed convex hull of a weakly compact subset of aBanach space is weakly compact.

The book ‘An introduction to Banach Space Theory’ by RobertE. Megginson, Springer 1998, describes these topologies, and afew others, particularly well.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Consider an open connected domain Ω ⊂ R3 representingvacuum.

Let E : Ω→ R3 be an electric field.

Path integrals of E between two points do not depend on thepath.

There is therefore a scalar field u : Ω→ R, the electricpotential, that satisfies ∇u = E .

The electric field is divergence-free and this implies∆u = ∇ · ∇u = 0.

Assume the boundary ∂Ω is a surface on which there iselectrical charge described by a function f : ∂Ω→ R.

The question is if there is an extension u of u so that u |∂Ω= fand ∆u = 0 in Ω, the celebrated classical Dirichlet problem.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Consider an open connected domain Ω ⊂ R3 representingvacuum.

Let E : Ω→ R3 be an electric field.

Path integrals of E between two points do not depend on thepath.

There is therefore a scalar field u : Ω→ R, the electricpotential, that satisfies ∇u = E .

The electric field is divergence-free and this implies∆u = ∇ · ∇u = 0.

Assume the boundary ∂Ω is a surface on which there iselectrical charge described by a function f : ∂Ω→ R.

The question is if there is an extension u of u so that u |∂Ω= fand ∆u = 0 in Ω, the celebrated classical Dirichlet problem.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Consider an open connected domain Ω ⊂ R3 representingvacuum.

Let E : Ω→ R3 be an electric field.

Path integrals of E between two points do not depend on thepath.

There is therefore a scalar field u : Ω→ R, the electricpotential, that satisfies ∇u = E .

The electric field is divergence-free and this implies∆u = ∇ · ∇u = 0.

Assume the boundary ∂Ω is a surface on which there iselectrical charge described by a function f : ∂Ω→ R.

The question is if there is an extension u of u so that u |∂Ω= fand ∆u = 0 in Ω, the celebrated classical Dirichlet problem.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Consider an open connected domain Ω ⊂ R3 representingvacuum.

Let E : Ω→ R3 be an electric field.

Path integrals of E between two points do not depend on thepath.

There is therefore a scalar field u : Ω→ R, the electricpotential, that satisfies ∇u = E .

The electric field is divergence-free and this implies∆u = ∇ · ∇u = 0.

Assume the boundary ∂Ω is a surface on which there iselectrical charge described by a function f : ∂Ω→ R.

The question is if there is an extension u of u so that u |∂Ω= fand ∆u = 0 in Ω, the celebrated classical Dirichlet problem.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Consider an open connected domain Ω ⊂ R3 representingvacuum.

Let E : Ω→ R3 be an electric field.

Path integrals of E between two points do not depend on thepath.

There is therefore a scalar field u : Ω→ R, the electricpotential, that satisfies ∇u = E .

The electric field is divergence-free and this implies∆u = ∇ · ∇u = 0.

Assume the boundary ∂Ω is a surface on which there iselectrical charge described by a function f : ∂Ω→ R.

The question is if there is an extension u of u so that u |∂Ω= fand ∆u = 0 in Ω, the celebrated classical Dirichlet problem.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Consider an open connected domain Ω ⊂ R3 representingvacuum.

Let E : Ω→ R3 be an electric field.

Path integrals of E between two points do not depend on thepath.

There is therefore a scalar field u : Ω→ R, the electricpotential, that satisfies ∇u = E .

The electric field is divergence-free and this implies∆u = ∇ · ∇u = 0.

Assume the boundary ∂Ω is a surface on which there iselectrical charge described by a function f : ∂Ω→ R.

The question is if there is an extension u of u so that u |∂Ω= fand ∆u = 0 in Ω, the celebrated classical Dirichlet problem.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Consider an open connected domain Ω ⊂ R3 representingvacuum.

Let E : Ω→ R3 be an electric field.

Path integrals of E between two points do not depend on thepath.

There is therefore a scalar field u : Ω→ R, the electricpotential, that satisfies ∇u = E .

The electric field is divergence-free and this implies∆u = ∇ · ∇u = 0.

Assume the boundary ∂Ω is a surface on which there iselectrical charge described by a function f : ∂Ω→ R.

The question is if there is an extension u of u so that u |∂Ω= fand ∆u = 0 in Ω, the celebrated classical Dirichlet problem.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The view that nature somehow operates optimally suggestslooking for a quantity to minimize.

A possibility is to integrate the scalar field |E |, or better yet|E |2.

The latter quantity is known as the energy of the electrical field.

The idea is to minimize∫

Ω |∇u|2 on some space of functions.

Euler’s scheme of calculating the derivative in this setting andequating it to zero will here produce Laplace’s equation.

Why is there a potential of minimal energy?

Create a reflexive Banach space that holds the functions u andestablish that the energy tends to infinity if ||u|| tends toinfinity.

Consider the greatest lower bound, which here is non-negative.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The view that nature somehow operates optimally suggestslooking for a quantity to minimize.

A possibility is to integrate the scalar field |E |, or better yet|E |2.

The latter quantity is known as the energy of the electrical field.

The idea is to minimize∫

Ω |∇u|2 on some space of functions.

Euler’s scheme of calculating the derivative in this setting andequating it to zero will here produce Laplace’s equation.

Why is there a potential of minimal energy?

Create a reflexive Banach space that holds the functions u andestablish that the energy tends to infinity if ||u|| tends toinfinity.

Consider the greatest lower bound, which here is non-negative.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The view that nature somehow operates optimally suggestslooking for a quantity to minimize.

A possibility is to integrate the scalar field |E |, or better yet|E |2.

The latter quantity is known as the energy of the electrical field.

The idea is to minimize∫

Ω |∇u|2 on some space of functions.

Euler’s scheme of calculating the derivative in this setting andequating it to zero will here produce Laplace’s equation.

Why is there a potential of minimal energy?

Create a reflexive Banach space that holds the functions u andestablish that the energy tends to infinity if ||u|| tends toinfinity.

Consider the greatest lower bound, which here is non-negative.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The view that nature somehow operates optimally suggestslooking for a quantity to minimize.

A possibility is to integrate the scalar field |E |, or better yet|E |2.

The latter quantity is known as the energy of the electrical field.

The idea is to minimize∫

Ω |∇u|2 on some space of functions.

Euler’s scheme of calculating the derivative in this setting andequating it to zero will here produce Laplace’s equation.

Why is there a potential of minimal energy?

Create a reflexive Banach space that holds the functions u andestablish that the energy tends to infinity if ||u|| tends toinfinity.

Consider the greatest lower bound, which here is non-negative.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The view that nature somehow operates optimally suggestslooking for a quantity to minimize.

A possibility is to integrate the scalar field |E |, or better yet|E |2.

The latter quantity is known as the energy of the electrical field.

The idea is to minimize∫

Ω |∇u|2 on some space of functions.

Euler’s scheme of calculating the derivative in this setting andequating it to zero will here produce Laplace’s equation.

Why is there a potential of minimal energy?

Create a reflexive Banach space that holds the functions u andestablish that the energy tends to infinity if ||u|| tends toinfinity.

Consider the greatest lower bound, which here is non-negative.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The view that nature somehow operates optimally suggestslooking for a quantity to minimize.

A possibility is to integrate the scalar field |E |, or better yet|E |2.

The latter quantity is known as the energy of the electrical field.

The idea is to minimize∫

Ω |∇u|2 on some space of functions.

Euler’s scheme of calculating the derivative in this setting andequating it to zero will here produce Laplace’s equation.

Why is there a potential of minimal energy?

Create a reflexive Banach space that holds the functions u andestablish that the energy tends to infinity if ||u|| tends toinfinity.

Consider the greatest lower bound, which here is non-negative.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The view that nature somehow operates optimally suggestslooking for a quantity to minimize.

A possibility is to integrate the scalar field |E |, or better yet|E |2.

The latter quantity is known as the energy of the electrical field.

The idea is to minimize∫

Ω |∇u|2 on some space of functions.

Euler’s scheme of calculating the derivative in this setting andequating it to zero will here produce Laplace’s equation.

Why is there a potential of minimal energy?

Create a reflexive Banach space that holds the functions u andestablish that the energy tends to infinity if ||u|| tends toinfinity.

Consider the greatest lower bound, which here is non-negative.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

The view that nature somehow operates optimally suggestslooking for a quantity to minimize.

A possibility is to integrate the scalar field |E |, or better yet|E |2.

The latter quantity is known as the energy of the electrical field.

The idea is to minimize∫

Ω |∇u|2 on some space of functions.

Euler’s scheme of calculating the derivative in this setting andequating it to zero will here produce Laplace’s equation.

Why is there a potential of minimal energy?

Create a reflexive Banach space that holds the functions u andestablish that the energy tends to infinity if ||u|| tends toinfinity.

Consider the greatest lower bound, which here is non-negative.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A minimum must satisfy Laplace’s equation.

Look, there is only one function that does that and also agreeswith the boundary values.

It must be the solution.

Weierstrass (1870): “Not true! Here is an example.”

But, what about Riemann’s results in complex analysis thatwere motivated by the Dirichlet principle?

Well, other proofs were found.

The general existence question is much harder and still thesubject of much research.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A minimum must satisfy Laplace’s equation.

Look, there is only one function that does that and also agreeswith the boundary values.

It must be the solution.

Weierstrass (1870): “Not true! Here is an example.”

But, what about Riemann’s results in complex analysis thatwere motivated by the Dirichlet principle?

Well, other proofs were found.

The general existence question is much harder and still thesubject of much research.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A minimum must satisfy Laplace’s equation.

Look, there is only one function that does that and also agreeswith the boundary values.

It must be the solution.

Weierstrass (1870): “Not true! Here is an example.”

But, what about Riemann’s results in complex analysis thatwere motivated by the Dirichlet principle?

Well, other proofs were found.

The general existence question is much harder and still thesubject of much research.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A minimum must satisfy Laplace’s equation.

Look, there is only one function that does that and also agreeswith the boundary values.

It must be the solution.

Weierstrass (1870): “Not true! Here is an example.”

But, what about Riemann’s results in complex analysis thatwere motivated by the Dirichlet principle?

Well, other proofs were found.

The general existence question is much harder and still thesubject of much research.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A minimum must satisfy Laplace’s equation.

Look, there is only one function that does that and also agreeswith the boundary values.

It must be the solution.

Weierstrass (1870): “Not true! Here is an example.”

But, what about Riemann’s results in complex analysis thatwere motivated by the Dirichlet principle?

Well, other proofs were found.

The general existence question is much harder and still thesubject of much research.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A minimum must satisfy Laplace’s equation.

Look, there is only one function that does that and also agreeswith the boundary values.

It must be the solution.

Weierstrass (1870): “Not true! Here is an example.”

But, what about Riemann’s results in complex analysis thatwere motivated by the Dirichlet principle?

Well, other proofs were found.

The general existence question is much harder and still thesubject of much research.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

A minimum must satisfy Laplace’s equation.

Look, there is only one function that does that and also agreeswith the boundary values.

It must be the solution.

Weierstrass (1870): “Not true! Here is an example.”

But, what about Riemann’s results in complex analysis thatwere motivated by the Dirichlet principle?

Well, other proofs were found.

The general existence question is much harder and still thesubject of much research.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Select a minimizing sequence uk with energy approaching thegreatest lower bound.

The sequence will be bounded.

Establish that the energy is sequentially weakly lowersemi-continuous.

Appeal to the weak compactness consequences and concludedthat there is some convergent subsequence.

The limit is the sought-after solution.

This approach is known as ‘the direct method in the calculus ofvariations’ and originated from Tonelli’s response to some ofthe question raised in Hilbert’s list of 23 problems (1900).

Of the three that specifically relate to the calculus of variations,the 19th and 20th are now consider resolved but the 23d is not.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Select a minimizing sequence uk with energy approaching thegreatest lower bound.

The sequence will be bounded.

Establish that the energy is sequentially weakly lowersemi-continuous.

Appeal to the weak compactness consequences and concludedthat there is some convergent subsequence.

The limit is the sought-after solution.

This approach is known as ‘the direct method in the calculus ofvariations’ and originated from Tonelli’s response to some ofthe question raised in Hilbert’s list of 23 problems (1900).

Of the three that specifically relate to the calculus of variations,the 19th and 20th are now consider resolved but the 23d is not.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Select a minimizing sequence uk with energy approaching thegreatest lower bound.

The sequence will be bounded.

Establish that the energy is sequentially weakly lowersemi-continuous.

Appeal to the weak compactness consequences and concludedthat there is some convergent subsequence.

The limit is the sought-after solution.

This approach is known as ‘the direct method in the calculus ofvariations’ and originated from Tonelli’s response to some ofthe question raised in Hilbert’s list of 23 problems (1900).

Of the three that specifically relate to the calculus of variations,the 19th and 20th are now consider resolved but the 23d is not.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Select a minimizing sequence uk with energy approaching thegreatest lower bound.

The sequence will be bounded.

Establish that the energy is sequentially weakly lowersemi-continuous.

Appeal to the weak compactness consequences and concludedthat there is some convergent subsequence.

The limit is the sought-after solution.

This approach is known as ‘the direct method in the calculus ofvariations’ and originated from Tonelli’s response to some ofthe question raised in Hilbert’s list of 23 problems (1900).

Of the three that specifically relate to the calculus of variations,the 19th and 20th are now consider resolved but the 23d is not.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Select a minimizing sequence uk with energy approaching thegreatest lower bound.

The sequence will be bounded.

Establish that the energy is sequentially weakly lowersemi-continuous.

Appeal to the weak compactness consequences and concludedthat there is some convergent subsequence.

The limit is the sought-after solution.

This approach is known as ‘the direct method in the calculus ofvariations’ and originated from Tonelli’s response to some ofthe question raised in Hilbert’s list of 23 problems (1900).

Of the three that specifically relate to the calculus of variations,the 19th and 20th are now consider resolved but the 23d is not.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Select a minimizing sequence uk with energy approaching thegreatest lower bound.

The sequence will be bounded.

Establish that the energy is sequentially weakly lowersemi-continuous.

Appeal to the weak compactness consequences and concludedthat there is some convergent subsequence.

The limit is the sought-after solution.

This approach is known as ‘the direct method in the calculus ofvariations’ and originated from Tonelli’s response to some ofthe question raised in Hilbert’s list of 23 problems (1900).

Of the three that specifically relate to the calculus of variations,the 19th and 20th are now consider resolved but the 23d is not.

Topology

Linner

Topology ininfinitedimensions.

Special topic.

Select a minimizing sequence uk with energy approaching thegreatest lower bound.

The sequence will be bounded.

Establish that the energy is sequentially weakly lowersemi-continuous.

Appeal to the weak compactness consequences and concludedthat there is some convergent subsequence.

The limit is the sought-after solution.

This approach is known as ‘the direct method in the calculus ofvariations’ and originated from Tonelli’s response to some ofthe question raised in Hilbert’s list of 23 problems (1900).

Of the three that specifically relate to the calculus of variations,the 19th and 20th are now consider resolved but the 23d is not.