QUADERNI

dell'Istituto di Matematica

FACOLTA' DI ECONOMIA

UNIVERSITA' DI MESSINA

On the coordinates

in an ultralinearly

independent family

DAVID CARFI' e CLARA GERMANA'

Dottori di Ricerca presso l'Università di Messina

no1 `2000

Via dei Verdi, 75 - 98122 MESSINA (Italy)

1

Quaderni dell'Istituto di Matematica

Facoltà di Economia

Università di Messina

On the coordinates

in an ultralinearly

independent family

DAVID CARFI' e CLARA GERMANA'

Dottori di Ricerca presso l'Università di Messina

no1 `2000

Via dei Verdi, 75 - 98122 MESSINA (Italy)

Rivista con referee

2

Quaderni dell'Istituto di Matematica

Facoltà di Economia

Università di Messina

Comitato Scienti�co

Prof. Maria Teresa Calapso - Università di Messina

Prof. Marcel Decuyper - Professore onorario dell' Università di Lille - Francia

Prof. Benedetto Matarazzo - Università di Catania

Prof. Lorenzo Peccati - Università Bocconi - Milano

Prof. Mircea Predeleanu - Universitè Paris 6 - Francia

Prof. Radu Rosca - Accademia Reale del Belgio

Prof. Ronald Rosseau - Katholieke Industriele Hogeschool Ostende - Belgio

Prof. Bernard Rouxel - Universitè de Brest - Francia

Prof. Gilbert Saporta - Conservatoire National des Arts et Métiers - Parigi

Prof. Anton Stefanescu - Università di Bucarest - Romania

Prof. Leopold Verstraelen - Katholieke Universiteit Leuven - Belgio

Lavoro presentato da:

Prof. Maria Teresa Calapso

Prof. Radu Rosca

3

On the coordinates in an ultralinearly

independent family

David Carfì

Faculty of Economics, University of Messina, Italy

Email: davidcar�@gmail.com

Clara Germanà

Faculty of Economics, University of Messina, Italy

Email: cgermana@unime.it

Dedicated to Professor E. Zeidler, with admiration and esteem.

Abstract

In this paper we study some aspects of new concepts introduced by D.Carfì; the new concepts are:

1) the coordinates' operator with respect to an ultralinearly indepen-dent family;

2) the product of two SL-families;3) the invertible family;4) the superposition of a family with respect to a family;5) the family of change from an ultrabasis to another ultrabasis.These new concepts permit the development of a generalization of

linear algebra in the space of tempered distributions and a more deeplystudy of some problems faced by linear algebra, as the theory of system,theory of decision, the optimal control, the quantum mechanics and so on.

Mathematics Subject Classification (1991): 46F10, 46F99, 47A05, 47N50,70A05, 70B05, 81P05, 81Q99.

Key words: Linear operator, tempered distribution, basis, quantum system,

state, linear superposition, subspace, generator, linear independence, contravariant

components, system of coordinates, matrix, expansion of an operator, resolution of

the identity.

Introduction

Recently, to study a generalization of the linear structure of the space oftempered distributions, D. Carfì has introduced some new concepts, that, astime goes by, have acquired a primary importance in the application to quantummechanics and computational economics.

1

In particular in this paper we give for the �rst time a rigorously version totwo of the most important theorem of the Dirac calculus: the �change of basistheorem� and the �resolution of the identity theorem�.

0. Preliminaries and notations

In this paper we shall use the following notations and concepts:

1) n,m are natural numbers;

2) Sn := S(Rn,C) is the Schwartz space, that is to say the set of all thesmooth functions (i.e. of class C∞) of Rn in C rapidly decreasing at in�nityand S(Rn,C) is the standard Schwartz topology on S(Rn,C);

3) µn is the Lebesgue measure in Rn; (·)(R,C) is the immersion of R in C and

if X is a non empty set (·)X is the identic function on X;

4) S ′n := S ′(Rn,C) is the space of tempered distributions from Rn to C, thatis the dual of the topological vector space (S(Rn,C),S(Rn,C)) i.e.

S ′(Rn,C) =(S(Rn,C),S(Rn,C))∗ = L(S(Rn,C),C),

where, if X and Y are two topological vector spaces, Hom(X,Y ) is the set of alllinear operator from X to Y and L(X,Y ) is the set of all linear and continuousoperator from X to Y ;

5) if a ∈ Rn, δa is the distribution of Dirac centered at a, i.e. the functional:

δa : S(Rn,C)→ C : φ 7→ φ (a)

where D(Rn,C) is the space of the smooth (of class C∞) with compact support;

6) we denote by s (Rm,S ′(Rn,C)) the space of all families is S ′(Rn,C) withRn as indices set, i.e. the set of all functions from Rm to S ′(Rn,C).

Moreover, let v be one of these families, for each p ∈ Rm, the distributionv(p) is denoted by vp.

The set s (Rm,S ′(Rn,C)) is a vector space with respect the following twostandard operations:

i) the addition

+ : s (Rm,S ′(Rn,C))2 → s (Rm,S ′(Rn,C)) : (v, w) 7→ v + w

where v + w is the family de�ned by

v + w : Rm → S ′(Rn,C) : p 7→ vp + wp,

i.e.(v + w) (p) = (v + w)p = vp + wp;

ii) the multiplication by scalars

· : C× s (Rm,S ′(Rn,C))→ s (Rm,S ′(Rn,C)) : (λ, v) 7→ λv

2

where λv is the family:

λv : Rm → S ′(Rn,C) : p 7→ λvp

i.e.(λv) (p) = (λv)p = λvp

in the sequel we shall denote s (Rm,S ′(Rn,C)) by X;

7) if U is an open subset of Rn, and f ∈ L1loc(U, C), then

〈f | = 〈f |n : D(U,C)→ C : g 7→∫U

fgdµn

is the regular distribution generated by f ; if g ∈ L1loc(U,C) and fg ∈ L1(U,C)

we put

〈f |g〉n := 〈g|n (f) := 〈f |n (g) :=∫U

fgdµn;

8) S(h,ω) = F(h,ω) is the (h, ω)-Fourier-Schwartz transformation (where h, ω ∈R 6= = R\{0}) i.e. the operator

S(h,ω) : S(Rn, C)→S(Rn,C),

such that, for all f ∈ S(Rn,C) and a ∈ Rn, one has

S(h,ω)(f)(a) =(1

h

)n ∫Rn

fe−iω(·|a)dµn,

where (· | ·) is the standard scalar product on Rn. Moreover, we recall thatS(h,ω) is an homeomorphism and, about its inverse, one has

S−(h,ω)(f)(a) =(|ω|h2π

)n ∫Rn

feiω(·|a)dµn = S(2π/(|ω|h),−ω)(f)(a),

i.e.

f(x) =

(|ω|h2π

)n ∫Rn

eiω(x|·)F(h,ω)(f)dµn = F(2π/(|ω|h),−ω)(F(h,ω)(f)

)(x).

With F(h,ω) we shall denote also the (h, ω)-Fourier transformation (where

h, ω ∈ R 6=) on the space of tempered distributions, i.e. the operator

F(h,ω) : S ′(Rn, C)→S ′(Rn,C),

such that, for all u ∈ S ′(Rn,C) and for any f ∈ S(Rn,C), one has

F(h,ω)(u)(f) = u(F(h,ω)(f)),

moreover, we recall that F(h,ω) is an homeomorphism and one has

F−(h,ω)(u) = F(2π/(|ω|h),−ω)(u);

3

9) moreover, we shall use the following de�nitions (see [10]):

De�nition 0.1 (family of tempered distributions of class S). Let T ∈s(Rm,S ′(Rn,C)) a family of distributions. One de�nes the family T familyof class S or S-family if, for each f ∈ S(Rn,C), the function T (f) of Rm in C,de�ned by T (f)(p) = Tp(f), for each p ∈ Rm, belongs to the space S(Rm,C)and the set of all these families is denoted by

S(Rm,S ′(Rn,C)) = S(m,S ′(Rn,C));

De�nition 0.2 (of operator generated by an S- family of tempered distribu-tions). Let T ∈ S(Rm,S ′(Rn,C)) be a family of tempered distributions of classS. One de�nes operator generated by the family T (or associated with T ) theoperator

T : S(Rn,C)→ S(Rm,C)

de�ned by T (f)(p) = Tp(f), for each f in S(Rn,C) and for each p in Rm, i.e.with the notations of the above de�nition, de�ned by T (f) = T (f), for each fin S(Rn,C);

De�nition 0.3 (family of tempered distributions of class SL). Let v ∈S (Rm,S ′(Rn,C)) , a family of distributions. One de�nes the family v family ofclass SL or SL-family if one has

v ∈ L (S(Rn,C),S(Rm,C)) .

The family of such systems is denoted by

SL (Rm,S ′(Rn,C)) = SL(m,S ′(Rn,C)).

10) In [10] we have state and prove the following theorems:

Theorem 0.1 (of structure). The set S (Rm,S ′(Rn,C)) is a subspace of thevector space (X,+, ·) .

Theorem 0.2 (of linear embedding). The application

(·)∧ : S (Rm,S ′ (Rn,C))→ Hom(S (Rn,C) ,S (Rm,C)) : v 7→ v

is an injective linear operator, thus one has

(v + w)∧= v + w

and (λv)∧= λv.

Theorem 0.3 (of structure). The set SL (Rm,S ′(Rn,C)) is a subspace of thevector space

(S (Rm,S ′(Rn,C)) ,+, ·) .

Theorem 0.4 (of isomorphism). The function

(·)∧ : SL (Rm,S ′ (Rn,C))→ L (S (Rn,C) ,S (Rm,C))

de�ned by (·)∧ (v) = v is an isomorphism.

4

11) In [9] one gives the following de�nitions:

De�nition 0.4 (linear superpositions of an SL-family). Let

a ∈ S ′(Rm,C)

andv ∈ SL (Rm,S ′(Rn,C)) .

One de�nes generalized linear combination of v with respect to (the systemof coe�cients) a or ultralinear combination of v with respect to (the systemof coe�cients) a, or linear superposition of v with respect to (the system ofcoe�cients) a, the distribution∫

Rm

av := a ◦ v : φ 7→ a (v(φ)) .

Moreover, if u ∈ S ′(Rn,C) and there exists an a ∈ S ′(Rm,C) such that

u =

∫Rm

av,

u is said an S ′-linear superposition of v. Finally, we de�ne linear superpositionof v the distribution ∫

Rm

v :=

∫Rm

⟨1(Rm,C)

∣∣ v,where

⟨1(Rm,C)

∣∣ is the regular distribution generated by the complex constantfunctional on Rm of value 1;

De�nition 0.5 (of S-ultralinear independence). Let v ∈ SL(Rm,S ′(Rn,C)).One de�nes v S-ultralinearly independent, if one has(

u ∈ S ′(Rm,C) ∧∫Rm

uv = 0S′(Rn,C)

)⇒ u = 0S′(Rm,C);

De�nition 0.6 (of generalized linear span). Let

v ∈ SL (Rm,S ′(Rn,C)) .

One de�nes S ′-ultralinear span of v, and it's denoted by Suspan (v), the set{u ∈ S ′(Rn,C) : ∃a ∈ S ′(Rm,C) : u =

∫Rm

av

};

De�nition 0.7 (system of S-ultragenerators). Let T ∈ SL (Rm,S ′(Rn,C)).T is de�ned system of S-ultragenerators for X ⊆ S ′(Rn,C) if and only if

Suspan (T ) = X;

De�nition 0.8 (of S-ultrabasis). Let v ∈ SL(Rm,S ′(Rn,C)) and let

X ⊆ S ′(Rn,C).

5

One de�nes v S-ultrabasis of X if it is S -ultralinearly independent, and onehas

Suspan(v) = X;

De�nition 0.9 (the system of contravariant components). Let

v ∈ SL(Rm,S ′(Rn,C))

be a S-ultralinearly independent family and

w ∈ Suspan(v).

The only tempered distribution a ∈ S ′(Rm,C) such that

w =

∫Rm

av

is denoted by[w|v]

and is called the system of contravariant components of w with respect to v orthe system of coordinates of w with respect to v.

1. The coordinates' operator with respect to an ultralinearlyindependent family

De�nition 1.1 (of coordinates' operator with respect to an ultra-linearly independent family). Let w ∈ SL(m,n) be an ultralinearly inde-pendent family. One de�nes coordinates' operator with respect to w thefollowing function:

[· | w] : Suspan(w)→ S ′m : u 7→ [u | w] . �

Example 1.1 (on the Dirac family). Let δ be the Dirac family (see [9]).For all u ∈ S ′n, one has

[u | δ] = u

and hence[· | δ] = (·)S′

n. 4

Example 1.2 (on the (h, ω)−Fourier family). Let f be the (h, ω)-Fourierfamily. For each u ∈ S ′n one has

[u | f ] = F−(h,ω)(u)

(see [9]), and hence[· | f ] = F−(h,ω). 4

Theorem 1.1 Let w ∈ SL(m,n) be an ultralinearly independent family.Then, one has

[· | w] ∈ Hom(Suspan(w),S ′m).

6

Proof. Let λ ∈ C and u, v ∈ Suspan(w), then one has

u+ λv =

∫Rm

[u | w]w + λ

∫Rm

[v | w]w =

=

∫Rm

([u | w] + λ [v | w])w,

and thus, one has

[u+ λv | w] = [u | w] + λ [v | w] . Q.E.D. �

Theorem 1.2 Let w ∈ SL(m,n) and λ ∈ C 6=.

Then, the following assertions hold1) if w is ultralinearly independent the family λw is ultralinearly inde-

pendent;2) Suspan(w) = Suspan(λw);3) if w is ultralinearly independent, for each u ∈ Suspan(w), one has

[u | λw] = (1/λ) [u | w].

Proof. 1) Let a ∈ S ′m be such that∫Rm

a(λw) = 0S′n,

one has

0S′n=

∫Rm

a(λw) = λ

∫Rm

aw,

thus ∫Rm

aw = 0S′n/λ = 0S′

n,

now because w is ultralinearly independent one has

a = 0S′n.

2) Let u ∈ Suspan(w). Then, there exists an a ∈ S ′m such that

u =

∫Rm

aw.

Now, one has

u =λ

λ

∫Rm

aw =

∫Rm

(aλ

)(λw) ,

so u ∈ Suspan(λw) and hence

Suspan(w) ⊆ Suspan(λw).

7

Let u ∈ Suspan(λw). Then, there exists an a ∈ S ′m such that

u =

∫Rm

a(λw).

Now, one has (see [9])

u =

∫Rm

(λa)w,

and hence u ∈ Suspan(w), so

Suspan(λw) ⊆ Suspan(w).

ConcludingSuspan(w) = Suspan(λw).

3) For any u ∈ S ′n, one has

u =

∫Rm

[u | w]w,

hence

λu = λ

∫Rm

[u | w]w =

∫Rm

[u | w] (λw) ,

concluding

u =1

λ

∫Rm

[u | w] (λw) =

=

∫Rm

(1

λ[u | w]

)(λw) =

=

∫Rm

[(1/λ)u | w] (λw) . Q.E.D. �

De�nition 1.2 (product of two SL-families). Let k ∈ N,

v ∈ SL(Rk,S ′ (Rm,C)

)and

w ∈ SL (Rm,S ′ (Rn,C))

be two families of distributions. One de�nes product of v by w the family

v · w : Rk → S ′ (Rn,C) : p 7→∫Rm

vpw,

it is also denoted by ∫Rm

vw,

so, for each p ∈ Rk, one has

(v · w)p =(∫

Rm

vw

)(p) =

∫Rm

vpw. �

8

It can be proved that the product of two SL-families is an SL-family andthat

(a · b)∧ = a ◦ b.

De�nition 1.3 (of invertible SL-family). Let a ∈ SL (Rn,S ′ (Rn,C)).a is called invertible if there exists a

b ∈ SL (Rn,S ′ (Rn,C))

such thata · b = b · a = δ. �

Remark 1.1 It's easy to prove that for each invertible family a ∈ SL(n, n)there exists only a b ∈ SL(n, n) such that

a · b = b · a = δ.

This family is denoted by a−, moreover it can be proved that the operatorgenerated by a is invertible and one has(

a−)∧

= (a)−.

In fact, for each φ ∈ Sn, one has[(a−)∧ ◦ a] (φ) (p) =

(a−)∧

(a(φ)) (p) =

=(a−)p(a(φ)) =

=

(∫Rn

(a−)pa

)(φ) =

=(a− · a

)p(φ) =

= δp (φ) =

= φ (p) ;

so (a−)∧ ◦ a = (·)Sn .

Analogously we havea ◦(a−)∧

= (·)Sn ,

and hence a is invertible and (a−)∧= (a)−. N

Theorem 1.3 Let v, w ∈ SL(n,S ′n) be two ultralinearly independent family.Then, v · w is ultralinearly independent. Moreover, if w is invertible one has

[u | w · v] =∫Rn

[u | v]w−.

9

Proof. Let a ∈ S ′n be such that∫Rn

a(v · w) = 0S′n,

one has

0S′n

=

∫Rn

a(v · w) =

= a ◦ (v · w)∧ =

= a ◦ (v ◦ w) == (a ◦ v) ◦ w,

because w is ultralinearly independent, one has

a ◦ v = 0S′n

and because v is ultralinearly independent, one has a = 0S′n, so v ·w is ultralin-

early independent.If w is invertible then w is invertible and one has

u =

∫Rn

[u | v] v =

= [u | v] ◦ v =

= [u | v] ◦ w− ◦ w ◦ v =

=([u | v] ◦ w−

)◦ (w ◦ v) =

= ([u | v] ◦ w−) ◦ (w · v)∧ =

=

∫Rn

([u | v] ◦ w−

)(w · v) ,

and hence

[u | w · v] = [u | v] ◦ w− =

= [u | v] ◦(w−)∧

=

=

∫Rn

[u | v]w−. Q.E.D. �

2. A change of basis theorem

De�nition 2.1 (superposition of a family with respect to a family).Let v ∈ s(k,S ′m) and w ∈ SL(m,S ′n). The family∫

Rm

vw : Rk → S ′n : p 7→∫Rm

vpw

is called the superposition of w with respect to v. �

10

Example 2.1 Let w ∈ SL(Rm,S ′n) be an ultralinearly independent familyand let v ∈ s(Rk,S ′n) be such that

vp ∈ Suspan (w)

for each p ∈ Rk. Then, for each p ∈ Rk, one has

vp =

∫Rm

[vp | w]w,

i.e.

v =

∫Rm

[v | w]w,

where [v | w] is the family de�ned by:

[v | w] : Rk → S ′m : p→ [vp | w] ,

in fact (∫Rm

[v | w]w)(p) =

∫Rm

[v | w]p w =

=

∫Rm

[vp | w]w =

= vp. 4

Remark 2.1 Obviously, in the conditions of the above de�nition, if v ∈SL(k,S ′m) one has ∫

Rm

vw = v · w,

and thus ∫Rm

vw ∈ SL(k,S ′n). N

Theorem 2.1 (SL-ultralinearity of the SL-ultralinear combinations).Let

a ∈ S ′k,

v ∈ SL(Rk,S ′ (Rm,C)

)and

w ∈ SL (Rm,S ′ (Rn,C))

be two families of distributions. Then, one has∫Rm

(∫Rk

av

)w =

∫Rk

a

(∫Rm

vw

)=

∫Rk

a(v · w).

11

Proof. One has(∫Rm

(∫Rk

av

)w

)(φ) =

(∫Rm

av

)(w (φ)) =

= a(v (w(φ))) =

= a ((v ◦ w) (φ)) == a (v · w)∧ (φ) =

=

∫Rk

a (v · w) (φ) =

=

∫Rk

a

(∫Rm

vw

)(φ) ,

this because one has

(v · w)∧ (φ)(p) = (v · w)p (φ) =

=

(∫Rm

vpw

)(φ) =

= vp (w(φ)) =

= v (w(φ)) (p) =

= (v ◦ w)(φ)(p). �

Notation 2.1 (the set of all S-ultrabases of a subspace). Let X ⊆ S ′nbe a subspace. In the following we shall use the notation:

BL(Rm, X) = {v ∈ SL(Rm,S ′n) : Im(v) ⊆ X and v is an ultrabasis for X}. �

De�nition 2.2 (the family of change). Let v ∈ BL(n,S ′n) and w ∈BL(k,S ′n) be two families of distributions. One de�nes family of change from

v to w the following family:

[w | v] : Rn → S ′n : p 7→ [wp | v]. �

Theorem 2.2 (on the change of basis). Let v, w ∈ BL(n,S ′n) be suchthat

[v | w] ∈ SL(n, n)and u ∈ S ′n. Then, one has

[u | w] =∫Rn

[u | v] [v | w] .

Proof. One has (see Example 2.1)

v =

∫Rn

[v | w]w,

12

now, applying the SL-ultralinearity of the SL -ultralinear combinations (theo-rem 2.1), one has

u =

∫Rn

[u | v] v =

=

∫Rn

[u | v](∫

Rn

[v | w]w)

=

=

∫Rn

(∫Rn

[u | v] [v | w])w,

and thus

[u | w] =∫Rn

[u | v] [v | w] . �

3. Some resolution of identity theorems

De�nition 3.1 (superposition of a family with respect to an oper-ator). Let X ⊆ S ′n be a subspace of S ′n, A ∈ Hom(X,S ′m) and

v ∈ SL (Rm,S ′ (Rn,C))

be a family of distributions. One de�nes superposition of v with respect to

A, the operator ∫Rm

Av : X → S ′n : u 7→∫Rm

A(u)v. �

Theorem 3.1 (the resolution of the identity). Let

v ∈ SL(Rn,S ′n)

be an ultralinearly independent family. Then, one has

v ∈ BL(n,S ′n)

if and only if one has

(·)S′n=

∫Rn

[· | v]v.

Proof. (⇒) If v ∈ BL(n,S ′n), then

Suspan(v) = S ′n

and thus, for all u ∈ S ′n one has

u =

∫Rn

[u | v]v

13

i.e.

(·)S′n=

∫Rn

[· | v]v.

(⇐) If

(·)S′n=

∫Rn

[· | v]v

then, for all u ∈ S ′n one has

u =

∫Rn

[u | v]v

and henceS ′n = Suspan(v)

and sov ∈ BL(n,S ′n). �

Theorem 3.2 (the general resolution of the identity). Let v ∈SL(m,S ′n) be an ultralinearly independent family and X ⊆ S ′n a subspace ofS ′n. Then,

v ∈ BL(Rm, X)

if and only if

(·)X =

∫Rm

[· | v]v.

Proof. (⇒) Ifv ∈ BL(Rm, X)

then, for all u ∈ X, one has

u =

∫Rm

[u | v]v,

i.e.

(·)X (u) =

(∫Rm

[· | v]v)(u)

and thus

(·)X =

∫Rm

[· | v]v.

(⇐) If

(·)X =

∫Rm

[· | v]v,

then, for each u ∈ X, one has

u = (·)X (u) =

(∫Rm

[· | v]v)(u) =

∫Rm

[u | v]v

and henceX = Suspan(v). �

14

References

[1] J. Barros-Neto. An Introduction to the theory of distributions. MarcelDekker, Inc. NewYork, 1973.

[2] D. Carfì. Quantum operators and their action on tempered distributions.Booklets of the Mathematics Institute of the Faculty of Economics, Uni-versity of Messina, (10):1�20, 1996. Available as Researchgate Paper athttps://www.researchgate.net/publication/210189140\_Quantum\

_operators\_and\_their\_action\_on\_tempered\_distributions.

[3] D. Carfì. Principles of a generalization of the linear algebra in S′n.Booklets of the Mathematics Institute of the Faculty of Economics,University of Messina, (4):1�14, 1997. Available as Researchgate Pa-per at https://www.researchgate.net/publication/210189138\

_Principles\_of\_a\_generalization\_of\_the\_linear\_algebra\

_in\_the\_spaces\_of\_tempered\_distributions.

[4] D. Carfì. A conformally invariant caracterization of constant mean cur-vature surfaces in 3-dimensional space forms. Researchgate Paper, pages1�15, 1998. https://dx.doi.org/10.13140/RG.2.1.2023.7526.

[5] D. Carfì. On Pseudo-Riemannian manifolds with Minkowski indexcarrying skew symmetric Killing vectors Field. Rendiconti del SeminarioMatematico di Messina, 5(series II):91�98, 1998. Available as ResearchgatePaper at https://www.researchgate.net/publication/210189136\

_On\_Pseudo-Riemannian\_manifolds\_with\_Minkowski\_index\

_carrying\_skew\_symmetric\_Killing\_vectors\_Field.

[6] D. Carfì. SL-ultralinear operators and some their applications. Book-lets of the Mathematics Institute of the Faculty of Economics, Uni-versity of Messina, pages 1�13, 1998. Available as ResearchgatePaper at https://www.researchgate.net/publication/210189135\

_SL-ultralinear\_operators\_and\_some\_their\_applications.

[7] D. Carfì. Skew Symmetric Killing Vector Fields on a ParakahelerianManifold. Rendiconti del Seminario Matematico di Messina, 7(se-ries II):117�124, 2000. Available as Researchgate Paper at https://

www.researchgate.net/publication/210189129\_Skew\_Symmetric\

_Killing\_Vector\_Fields\_on\_a\_parakahelerian\_Manifold.

[8] D. Carfì. SL-ultralinear operators. Annals of Economic Fac-ulty, University of Messina, 38:195�214, 2000. Available as Re-searchgate Paper at https://www.researchgate.net/publication/

210189128\_SL-ultralinear\_operators.

[9] D. Carfì. S-Ultralinear Algebra in the space of tempered distributions.AAPP | Physical, Mathematical, and Natural Sciences, 2001. (in print).

[10] D. Carfì and C. Germanà. S-nets in the space of tempered distribu-tions and generated operators. Rendiconti del Seminario Matematico di

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Messina, 6(series II):113�124, 1999. Available as Researchgate Paper athttps://www.researchgate.net/publication/267462823\_S-nets\

_in\_the\_space\_of\_tempered\_distributions\_and\_generated\

_operators.

[11] D Carfì and C.. Germanà. The space of multipliers of S′ and the S-familiesof tempered distributions. Booklets of the Mathematics Institute of theFaculty of Economics, University of Messina, (5):1�15, 1999. Available asResearchgate Paper at https://www.researchgate.net/publication/

210189132\_The\_space\_of\_multipliers\_of\_S\%27\_and\_the\

_S-families\_of\_tempered\_distributions.

[12] D. Carfì and C. Germanà. S′-operators and generated families.Booklets of the Mathematics Institute of the Faculty of Economics,University of Messina, (2):1�16, 2000. Available as ResearchgatePaper at https://www.researchgate.net/publication/210189126\_S\%27-operators\_and\_generated\_families.

[13] D. Carfì and C. Germanà. The coordinate operator in SL-ultralinear alge-bra. Booklets of the Mathematics Institute of the Faculty of Economics,University of Messina, (4):1�13, 2000. Available as Researchgate Pa-per at https://www.researchgate.net/publication/210189125\_The\_coordinate\_operator\_in\_SL-ultralinear\_algebra.

[14] D. Carfì and E. Musso. T-transformations of Willmore isothermic surfaces.Rendiconti del Seminario Matematico di Messina, (series II):69�86,2000. Conference in honor of P. Calapso. Available as ResearchgatePaper at https://www.researchgate.net/publication/266991245\

_T-transformations\_of\_Willmore\_isothermic\_surfaces.

[15] J. B. Conway. A course in functional analysis. Springer Verlag, 1985.

[16] G. Gilardi. Analisi 3. McGraw-Hill, 1994.

[17] I.M. Guelfand and G.E. Chilov. Les Distributions (Tome 2). Dunod, Paris,1964.

[18] S. Kesavan. Topics in Functional analysis and applications. Wiley EasternLimited, 1989.

[19] S. Lang. Real and functional analisys. Springer Verlag, 1993.

[20] C. Miranda. Istituzioni di Analisi funzionale. Unione Matematica Italiana,1978.

[21] S. Roman. Advanced Linear Algebra. Springer-Verlag, New York, 1992.

[22] W. Rudin. Real and complex analysis. Mc Graw-Hill, 1966.

[23] W. Rudin. Functional analysis. Mc Graw-Hill, 1977.

[24] L. Schwartz. Théorie des Distributions. Hermann, Paris, 1978.

[25] E. Zeidler. Nonlinear Functional Analysis and its applications, volume IV.Springer Verlag, 1985.

[26] E. Zeidler. Applied Functional Analysis, volume I. Springer Verlag, 1995.

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