the krein{milman theorem - uppsala universitygaidash/presentations/presentation_s_pettersson.pdf ·...
TRANSCRIPT
![Page 1: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/1.jpg)
The Krein–Milman TheoremA Project in Functional Analysis
Samuel Pettersson
November 29, 2016
![Page 2: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/2.jpg)
Outline
1. An informal example
2. Extreme points
3. The Krein–Milman theorem
4. An application
![Page 3: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/3.jpg)
Outline
1. An informal example
2. Extreme points
3. The Krein–Milman theorem
4. An application
![Page 4: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/4.jpg)
Outline
1. An informal example
2. Extreme points
3. The Krein–Milman theorem
4. An application
![Page 5: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/5.jpg)
Outline
1. An informal example
2. Extreme points
3. The Krein–Milman theorem
4. An application
![Page 6: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/6.jpg)
Outline
1. An informal example
2. Extreme points
3. The Krein–Milman theorem
4. An application
![Page 7: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/7.jpg)
Convex sets and their “corners”
ObservationSome convex sets are the convex hulls of their “corners”.
![Page 8: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/8.jpg)
Convex sets and their “corners”
ObservationSome convex sets are the convex hulls of their “corners”.
‖x‖1≤ 1
![Page 9: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/9.jpg)
Convex sets and their “corners”
ObservationSome convex sets are the convex hulls of their “corners”.
‖x‖1≤ 1
![Page 10: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/10.jpg)
Convex sets and their “corners”
ObservationSome convex sets are the convex hulls of their “corners”.
‖x‖1≤ 1 ‖x‖2≤ 1
![Page 11: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/11.jpg)
Convex sets and their “corners”
ObservationSome convex sets are the convex hulls of their “corners”.
‖x‖1≤ 1 ‖x‖2≤ 1
![Page 12: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/12.jpg)
Convex sets and their “corners”
ObservationSome convex sets are the convex hulls of their “corners”.
‖x‖1≤ 1 ‖x‖2≤ 1 ‖x‖∞≤ 1
![Page 13: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/13.jpg)
Convex sets and their “corners”
ObservationSome convex sets are the convex hulls of their “corners”.
‖x‖1≤ 1 ‖x‖2≤ 1 ‖x‖∞≤ 1
![Page 14: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/14.jpg)
Convex sets and their “corners”
ObservationSome convex sets are not the convex hulls of their “corners”.
x1, x2 ≥ 0
![Page 15: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/15.jpg)
Convex sets and their “corners”
ObservationSome convex sets are not the convex hulls of their “corners”.
x1, x2 ≥ 0
![Page 16: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/16.jpg)
Convex sets and their “corners”
ObservationSome convex sets are not the convex hulls of their “corners”.
x1, x2 ≥ 0
![Page 17: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/17.jpg)
Convex sets and their “corners”
ObservationSome convex sets are not the convex hulls of their “corners”.
x1, x2 ≥ 0 ‖x‖∞< 1
![Page 18: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/18.jpg)
Objectives
I Formalize the notion of a corner of a convex set (extremepoint).
I Find a sufficient condition for a convex set to be the closedconvex hull of its extreme points (Krein–Milman).
![Page 19: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/19.jpg)
Objectives
I Formalize the notion of a corner of a convex set (extremepoint).
I Find a sufficient condition for a convex set to be the closedconvex hull of its extreme points (Krein–Milman).
![Page 20: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/20.jpg)
Objectives
I Formalize the notion of a corner of a convex set (extremepoint).
I Find a sufficient condition for a convex set to be the closedconvex hull of its extreme points (Krein–Milman).
![Page 21: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/21.jpg)
Outline
1. An informal example
2. Extreme points
3. The Krein–Milman theorem
4. An application
![Page 22: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/22.jpg)
Definition of an extreme point
DefinitionAn extreme point of a convex set K ⊆ E in a vector space E is apoint z ∈ K not in the interior of any line segment in K :
z 6= (1− t)x + ty , ∀t ∈ (0, 1), ∀x , y ∈ K , x 6= y
RemarkIn a normed space,
I interior points are never extremal
I boundary points may be extremal
I boundary points (inside the set) of strictly convex sets arealways extremal
![Page 23: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/23.jpg)
Definition of an extreme point
DefinitionAn extreme point of a convex set K ⊆ E in a vector space E is apoint z ∈ K not in the interior of any line segment in K :
z 6= (1− t)x + ty , ∀t ∈ (0, 1), ∀x , y ∈ K , x 6= y
RemarkIn a normed space,
I interior points are never extremal
I boundary points may be extremal
I boundary points (inside the set) of strictly convex sets arealways extremal
![Page 24: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/24.jpg)
Definition of an extreme point
DefinitionAn extreme point of a convex set K ⊆ E in a vector space E is apoint z ∈ K not in the interior of any line segment in K :
z 6= (1− t)x + ty , ∀t ∈ (0, 1), ∀x , y ∈ K , x 6= y
RemarkIn a normed space,
I interior points are never extremal
I boundary points may be extremal
I boundary points (inside the set) of strictly convex sets arealways extremal
![Page 25: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/25.jpg)
Definition of an extreme point
DefinitionAn extreme point of a convex set K ⊆ E in a vector space E is apoint z ∈ K not in the interior of any line segment in K :
z 6= (1− t)x + ty , ∀t ∈ (0, 1), ∀x , y ∈ K , x 6= y
RemarkIn a normed space,
I interior points are never extremal
I boundary points may be extremal
I boundary points (inside the set) of strictly convex sets arealways extremal
![Page 26: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/26.jpg)
Definition of an extreme point
DefinitionAn extreme point of a convex set K ⊆ E in a vector space E is apoint z ∈ K not in the interior of any line segment in K :
z 6= (1− t)x + ty , ∀t ∈ (0, 1), ∀x , y ∈ K , x 6= y
RemarkIn a normed space,
I interior points are never extremal
I boundary points may be extremal
I boundary points (inside the set) of strictly convex sets arealways extremal
![Page 27: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/27.jpg)
Definition of an extreme point
DefinitionAn extreme point of a convex set K ⊆ E in a vector space E is apoint z ∈ K not in the interior of any line segment in K :
z 6= (1− t)x + ty , ∀t ∈ (0, 1), ∀x , y ∈ K , x 6= y
RemarkIn a normed space,
I interior points are never extremal
I boundary points may be extremal
I boundary points (inside the set) of strictly convex sets arealways extremal
![Page 28: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/28.jpg)
Examples of extreme points
Extreme points of the closed unit ball:
Space Extreme points
`1 ±ei = (0, . . . , 0,
ith term︷︸︸︷±1 , 0, . . . )
`p (1 < p < ∞) Entire unit sphere
`∞ (±1,±1, . . . )
![Page 29: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/29.jpg)
Examples of extreme points
Extreme points of the closed unit ball:
Space Extreme points
`1 ±ei = (0, . . . , 0,
ith term︷︸︸︷±1 , 0, . . . )
`p (1 < p < ∞) Entire unit sphere
`∞ (±1,±1, . . . )
![Page 30: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/30.jpg)
Examples of extreme points
Extreme points of the closed unit ball:
Space Extreme points
`1
±ei = (0, . . . , 0,
ith term︷︸︸︷±1 , 0, . . . )
`p (1 < p < ∞) Entire unit sphere
`∞ (±1,±1, . . . )
![Page 31: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/31.jpg)
Examples of extreme points
Extreme points of the closed unit ball:
Space Extreme points
`1 ±ei = (0, . . . , 0,
ith term︷︸︸︷±1 , 0, . . . )
`p (1 < p < ∞) Entire unit sphere
`∞ (±1,±1, . . . )
![Page 32: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/32.jpg)
Examples of extreme points
Extreme points of the closed unit ball:
Space Extreme points
`1 ±ei = (0, . . . , 0,
ith term︷︸︸︷±1 , 0, . . . )
`p (1 < p < ∞)
Entire unit sphere
`∞ (±1,±1, . . . )
![Page 33: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/33.jpg)
Examples of extreme points
Extreme points of the closed unit ball:
Space Extreme points
`1 ±ei = (0, . . . , 0,
ith term︷︸︸︷±1 , 0, . . . )
`p (1 < p < ∞) Entire unit sphere
`∞ (±1,±1, . . . )
![Page 34: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/34.jpg)
Examples of extreme points
Extreme points of the closed unit ball:
Space Extreme points
`1 ±ei = (0, . . . , 0,
ith term︷︸︸︷±1 , 0, . . . )
`p (1 < p < ∞) Entire unit sphere
`∞
(±1,±1, . . . )
![Page 35: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/35.jpg)
Examples of extreme points
Extreme points of the closed unit ball:
Space Extreme points
`1 ±ei = (0, . . . , 0,
ith term︷︸︸︷±1 , 0, . . . )
`p (1 < p < ∞) Entire unit sphere
`∞ (±1,±1, . . . )
![Page 36: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/36.jpg)
Outline
1. An informal example
2. Extreme points
3. The Krein–Milman theorem
4. An application
![Page 37: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/37.jpg)
Statement of Krein–Milman
Theorem (Krein–Milman)
A compact convex set K ⊆ E in a normed space coincides with theclosed convex hull of its extreme points:
K = conv(extK )
Reminder
convA := convA
= the smallest closed and convex set containing A.
![Page 38: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/38.jpg)
Statement of Krein–Milman
Theorem (Krein–Milman)
A compact convex set K ⊆ E in a normed space coincides with theclosed convex hull of its extreme points:
K = conv(extK )
Reminder
convA := convA
= the smallest closed and convex set containing A.
![Page 39: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/39.jpg)
Preparation for the proof: Extreme sets
DefinitionGiven a compact convex set K ⊆ E in a normed space, an extremeset is a subset M ⊆ K that is
I non-empty
I closed
I such that any line segment in K whose interior intersects Mhas endpoints in M:
∃t ∈ (0, 1) : (1− t)x + ty ∈ M =⇒ x , y ∈ M, ∀x , y ∈ K
![Page 40: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/40.jpg)
Preparation for the proof: Extreme sets
DefinitionGiven a compact convex set K ⊆ E in a normed space, an extremeset is a subset M ⊆ K that is
I non-empty
I closed
I such that any line segment in K whose interior intersects Mhas endpoints in M:
∃t ∈ (0, 1) : (1− t)x + ty ∈ M =⇒ x , y ∈ M, ∀x , y ∈ K
![Page 41: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/41.jpg)
Preparation for the proof: Extreme sets
DefinitionGiven a compact convex set K ⊆ E in a normed space, an extremeset is a subset M ⊆ K that is
I non-empty
I closed
I such that any line segment in K whose interior intersects Mhas endpoints in M:
∃t ∈ (0, 1) : (1− t)x + ty ∈ M =⇒ x , y ∈ M, ∀x , y ∈ K
![Page 42: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/42.jpg)
Preparation for the proof: Extreme sets
DefinitionGiven a compact convex set K ⊆ E in a normed space, an extremeset is a subset M ⊆ K that is
I non-empty
I closed
I such that any line segment in K whose interior intersects Mhas endpoints in M:
∃t ∈ (0, 1) : (1− t)x + ty ∈ M =⇒ x , y ∈ M, ∀x , y ∈ K
![Page 43: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/43.jpg)
Preparation for the proof: Extreme sets
DefinitionGiven a compact convex set K ⊆ E in a normed space, an extremeset is a subset M ⊆ K that is
I non-empty
I closed
I such that any line segment in K whose interior intersects Mhas endpoints in M:
∃t ∈ (0, 1) : (1− t)x + ty ∈ M =⇒ x , y ∈ M, ∀x , y ∈ K
![Page 44: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/44.jpg)
Preparation for the proof: Extreme sets
LemmaFor A ⊆ K an extreme set and f ∈ E ?,
B := {x ∈ A : 〈f , x〉 = maxy∈A
〈f , y〉}
= {maxima of f on A}
is an extreme subset of K .
Proposition
Every extreme set A ⊆ K contains an extreme point of K .
Proof.Use Zorn’s lemma and the above lemma (details omitted).
![Page 45: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/45.jpg)
Preparation for the proof: Extreme sets
LemmaFor A ⊆ K an extreme set and f ∈ E ?,
B := {x ∈ A : 〈f , x〉 = maxy∈A
〈f , y〉}
= {maxima of f on A}
is an extreme subset of K .
Proposition
Every extreme set A ⊆ K contains an extreme point of K .
Proof.Use Zorn’s lemma and the above lemma (details omitted).
![Page 46: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/46.jpg)
Preparation for the proof: Extreme sets
LemmaFor A ⊆ K an extreme set and f ∈ E ?,
B := {x ∈ A : 〈f , x〉 = maxy∈A
〈f , y〉}
= {maxima of f on A}
is an extreme subset of K .
Proposition
Every extreme set A ⊆ K contains an extreme point of K .
Proof.Use Zorn’s lemma and the above lemma (details omitted).
![Page 47: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/47.jpg)
Preparation for the proof: Extreme sets
LemmaFor A ⊆ K an extreme set and f ∈ E ?,
B := {x ∈ A : 〈f , x〉 = maxy∈A
〈f , y〉}
= {maxima of f on A}
is an extreme subset of K .
Proposition
Every extreme set A ⊆ K contains an extreme point of K .
Proof.Use Zorn’s lemma and the above lemma (details omitted).
![Page 48: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/48.jpg)
Preparation for the proof: Extreme sets
LemmaFor A ⊆ K an extreme set and f ∈ E ?,
B := {x ∈ A : 〈f , x〉 = maxy∈A
〈f , y〉}
= {maxima of f on A}
is an extreme subset of K .
Proposition
Every extreme set A ⊆ K contains an extreme point of K .
Proof.Use Zorn’s lemma and the above lemma (details omitted).
![Page 49: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/49.jpg)
Reminder
Theorem (Krein–Milman)
A compact convex set K ⊆ E in a normed space coincides with theclosed convex hull of its extreme points:
K = conv(extK )
![Page 50: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/50.jpg)
Proof of Krein–Milman
For conv(extK ) ⊆ K ,
K compact, convex, and K ⊇ extK
=⇒ K closed, convex, and K ⊇ extK
=⇒ conv(extK ) ⊆ K
![Page 51: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/51.jpg)
Proof of Krein–Milman
For conv(extK ) ⊆ K ,
K compact, convex, and K ⊇ extK
=⇒ K closed, convex, and K ⊇ extK
=⇒ conv(extK ) ⊆ K
![Page 52: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/52.jpg)
Proof of Krein–Milman
For conv(extK ) ⊆ K ,
K compact, convex, and K ⊇ extK
=⇒ K closed, convex, and K ⊇ extK
=⇒ conv(extK ) ⊆ K
![Page 53: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/53.jpg)
Proof of Krein–Milman
For conv(extK ) ⊆ K ,
K compact, convex, and K ⊇ extK
=⇒ K closed, convex, and K ⊇ extK
=⇒ conv(extK ) ⊆ K
![Page 54: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/54.jpg)
Proof of Krein–Milman
For K ⊆ conv(extK ),
K = ∅ =⇒ K ⊆ conv(extK )
Otherwise, argue by contradiction:
∃x ∈ K \ conv(extK )
=⇒ ∃f ∈ E ? : f (conv(extK )) < f (x) (Hahn–Banach)
=⇒ ∃f ∈ E ? : f (extK ) < f (x)
=⇒ ∃B ⊆ K extreme set without extreme points (Lemma)
=⇒ Contradiction! (Proposition)
Hence, K ⊆ conv(extK )
![Page 55: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/55.jpg)
Proof of Krein–Milman
For K ⊆ conv(extK ),
K = ∅ =⇒ K ⊆ conv(extK )
Otherwise, argue by contradiction:
∃x ∈ K \ conv(extK )
=⇒ ∃f ∈ E ? : f (conv(extK )) < f (x) (Hahn–Banach)
=⇒ ∃f ∈ E ? : f (extK ) < f (x)
=⇒ ∃B ⊆ K extreme set without extreme points (Lemma)
=⇒ Contradiction! (Proposition)
Hence, K ⊆ conv(extK )
![Page 56: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/56.jpg)
Proof of Krein–Milman
For K ⊆ conv(extK ),
K = ∅ =⇒ K ⊆ conv(extK )
Otherwise, argue by contradiction:
∃x ∈ K \ conv(extK )
=⇒ ∃f ∈ E ? : f (conv(extK )) < f (x) (Hahn–Banach)
=⇒ ∃f ∈ E ? : f (extK ) < f (x)
=⇒ ∃B ⊆ K extreme set without extreme points (Lemma)
=⇒ Contradiction! (Proposition)
Hence, K ⊆ conv(extK )
![Page 57: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/57.jpg)
Proof of Krein–Milman
For K ⊆ conv(extK ),
K = ∅ =⇒ K ⊆ conv(extK )
Otherwise, argue by contradiction:
∃x ∈ K \ conv(extK )
=⇒ ∃f ∈ E ? : f (conv(extK )) < f (x) (Hahn–Banach)
=⇒ ∃f ∈ E ? : f (extK ) < f (x)
=⇒ ∃B ⊆ K extreme set without extreme points (Lemma)
=⇒ Contradiction! (Proposition)
Hence, K ⊆ conv(extK )
![Page 58: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/58.jpg)
Proof of Krein–Milman
For K ⊆ conv(extK ),
K = ∅ =⇒ K ⊆ conv(extK )
Otherwise, argue by contradiction:
∃x ∈ K \ conv(extK )
=⇒ ∃f ∈ E ? : f (conv(extK )) < f (x) (Hahn–Banach)
=⇒ ∃f ∈ E ? : f (extK ) < f (x)
=⇒ ∃B ⊆ K extreme set without extreme points (Lemma)
=⇒ Contradiction! (Proposition)
Hence, K ⊆ conv(extK )
![Page 59: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/59.jpg)
Proof of Krein–Milman
For K ⊆ conv(extK ),
K = ∅ =⇒ K ⊆ conv(extK )
Otherwise, argue by contradiction:
∃x ∈ K \ conv(extK )
=⇒ ∃f ∈ E ? : f (conv(extK )) < f (x) (Hahn–Banach)
=⇒ ∃f ∈ E ? : f (extK ) < f (x)
=⇒ ∃B ⊆ K extreme set without extreme points (Lemma)
=⇒ Contradiction! (Proposition)
Hence, K ⊆ conv(extK )
![Page 60: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/60.jpg)
Proof of Krein–Milman
For K ⊆ conv(extK ),
K = ∅ =⇒ K ⊆ conv(extK )
Otherwise, argue by contradiction:
∃x ∈ K \ conv(extK )
=⇒ ∃f ∈ E ? : f (conv(extK )) < f (x) (Hahn–Banach)
=⇒ ∃f ∈ E ? : f (extK ) < f (x)
=⇒ ∃B ⊆ K extreme set without extreme points (Lemma)
=⇒ Contradiction! (Proposition)
Hence, K ⊆ conv(extK )
![Page 61: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/61.jpg)
Proof of Krein–Milman
For K ⊆ conv(extK ),
K = ∅ =⇒ K ⊆ conv(extK )
Otherwise, argue by contradiction:
∃x ∈ K \ conv(extK )
=⇒ ∃f ∈ E ? : f (conv(extK )) < f (x) (Hahn–Banach)
=⇒ ∃f ∈ E ? : f (extK ) < f (x)
=⇒ ∃B ⊆ K extreme set without extreme points (Lemma)
=⇒ Contradiction! (Proposition)
Hence, K ⊆ conv(extK )
![Page 62: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/62.jpg)
Proof of Krein–Milman
For K ⊆ conv(extK ),
K = ∅ =⇒ K ⊆ conv(extK )
Otherwise, argue by contradiction:
∃x ∈ K \ conv(extK )
=⇒ ∃f ∈ E ? : f (conv(extK )) < f (x) (Hahn–Banach)
=⇒ ∃f ∈ E ? : f (extK ) < f (x)
=⇒ ∃B ⊆ K extreme set without extreme points (Lemma)
=⇒ Contradiction! (Proposition)
Hence, K ⊆ conv(extK )
![Page 63: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/63.jpg)
Outline
1. An informal example
2. Extreme points
3. The Krein–Milman theorem
4. An application
![Page 64: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/64.jpg)
Not dual spaces
Example
c0 ⊆ `∞ and L1(R) are not dual spaces.
Proposition
The closed unit ball BE? has an extreme point.
Proof.
BE? is weakly? compact (Banach–Alaoglu–Bourbaki)
=⇒ BE? = conv(extBE?) (generalized Krein–Milman)
=⇒ BE? has an extreme point
![Page 65: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/65.jpg)
Not dual spaces
Example
c0 ⊆ `∞ and L1(R) are not dual spaces.
Proposition
The closed unit ball BE? has an extreme point.
Proof.
BE? is weakly? compact (Banach–Alaoglu–Bourbaki)
=⇒ BE? = conv(extBE?) (generalized Krein–Milman)
=⇒ BE? has an extreme point
![Page 66: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/66.jpg)
Not dual spaces
Example
c0 ⊆ `∞ and L1(R) are not dual spaces.
Proposition
The closed unit ball BE? has an extreme point.
Proof.
BE? is weakly? compact (Banach–Alaoglu–Bourbaki)
=⇒ BE? = conv(extBE?) (generalized Krein–Milman)
=⇒ BE? has an extreme point
![Page 67: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/67.jpg)
Not dual spaces
Example
c0 ⊆ `∞ and L1(R) are not dual spaces.
Proposition
The closed unit ball BE? has an extreme point.
Proof.
BE? is weakly? compact (Banach–Alaoglu–Bourbaki)
=⇒ BE? = conv(extBE?) (generalized Krein–Milman)
=⇒ BE? has an extreme point
![Page 68: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/68.jpg)
Not dual spaces
Example
c0 ⊆ `∞ and L1(R) are not dual spaces.
Proposition
The closed unit ball BE? has an extreme point.
Proof.
BE? is weakly? compact (Banach–Alaoglu–Bourbaki)
=⇒ BE? = conv(extBE?) (generalized Krein–Milman)
=⇒ BE? has an extreme point
![Page 69: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/69.jpg)
Not dual spaces
Example
c0 ⊆ `∞ and L1(R) are not dual spaces.
Proposition
The closed unit ball BE? has an extreme point.
Proof.
BE? is weakly? compact (Banach–Alaoglu–Bourbaki)
=⇒ BE? = conv(extBE?) (generalized Krein–Milman)
=⇒ BE? has an extreme point
![Page 70: The Krein{Milman Theorem - Uppsala Universitygaidash/Presentations/Presentation_S_Pettersson.pdf · The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November](https://reader033.vdocuments.mx/reader033/viewer/2022041821/5e5de3c7b6a0352bfe08730c/html5/thumbnails/70.jpg)
Not dual spaces
Example
c0 ⊆ `∞ and L1(R) are not dual spaces.
Proposition
The closed unit ball BE? has an extreme point.
Proof.
BE? is weakly? compact (Banach–Alaoglu–Bourbaki)
=⇒ BE? = conv(extBE?) (generalized Krein–Milman)
=⇒ BE? has an extreme point