operators and generated families

QUADERNI dell'Istituto di Matematica FACOLTA' DI ECONOMIA UNIVERSITA' DI MESSINA

S 0-operators and generated families

DAVID CARFI' e CLARA GERMANA' Dottori di Ricerca presso l'Università di Messina

no 2 `2000

Via dei Verdi, 75 - 98122 MESSINA (Italy)

1

Quaderni dell'Istituto di Matematica

Facoltà di Economia Università di Messina

S 0-operators and generated families

DAVID CARFI' e CLARA GERMANA' Dottori di Ricerca presso l'Università di Messina

no 2 `2000

Via dei Verdi, 75 - 98122 MESSINA (Italy)

Rivista con referee

2

Quaderni dell'Istituto di Matematica

Facoltà di Economia Università di Messina

Comitato Scientico

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

o¨

¨e

Prof. Ronald Rosseau - Kath lieke Industri le 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

o¨

Prof. Leopold Verstraelen - Kath lieke Universiteit Leuven - Belgio

Lavoro presentato da: Prof. Maria Teresa Calapso Prof. Radu Rosca

3

S 0-operators

and generated families

David Carfì

Department of Mathematics, University of Messina, Italy Email: davidcar@gmail.com

Clara Germanà

Faculty of Economics, University of Messina, Italy Email: [email protected] Dedicated to Professor Radu Rosca, with admiration and esteem.

Abstract

The aim of this paper is to study some new concepts introduced by D.Carfì to generalize the linear structure of the space of tempered distributions. The new concepts are: 1) operator of class S 0 or S 0 -operator; 2) family generated by an S 0 -operator; 3) superposition of a family of tempered distributions of class SL(see [9]) with respect to an operator; 0 4) image of a family of distributions under an operator A : Sn0 → Sm . These new concepts permit the development of a generalization of linear algebra in the space of tempered distributions and a more deeply study 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, representation of an operator.

0.

Preliminaries and notations

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

1)

n, m

2)

Sn := S(Rn , C)

are natural numbers;

smooth functions (i.e.

is the Schwartz space, that is to say the set of all the of class

C ∞)

of

1

Rn

in

C

rapidly decreasing at innity

and

S(Rn ,C)

3)

µn

is the standard Schwartz topology on

is the Lebesgue measure in

X

(·)X

is a non empty set

S(Rn , C);

Rn ; (·) = (·)(R,C)

is the immersion of

is the identic function on

R

in

X;

C

and if

is

Sn0 := S 0 (Rn , C) is the space of tempered distributions from Rn n the dual of the topological vector space (S(R , C), S(Rn ,C) ) i.e. 4)

to

C, that

S 0 (Rn , C) =(S(Rn , C), S(Rn ,C) )∗ = L(Sn , C), X

where, if

and

Y

operators from 5) if

Hom(X, Y ) is the set of all L(X, Y ) is the set of all linear and continuous

are two topological vector spaces,

linear operators from

X

to

a ∈ Rn , δ a

X to Y Y;

and

Sn

is the distribution of Dirac on

centered at

a,

i.e.

the

with

Rm

functional:

δa : Sn → C : φ 7→ φ (a) ; s (R , Sn0 ) = s(m, n) the space of all families in Sn0 m 0 the set of all functions from R to Sn . m

6) we denote by as indices set, i.e. Moreover, let

v(p)

v

be one of these families, for each

the distribution

vp .

is denoted by

s (Rm , Sn0 )

The set

p ∈ Rm ,

is a vector space with respect the following two standard

operations: i) the addition

2

+ : s (Rm , Sn0 ) → s (Rm , Sn0 ) : (v, w) 7→ v + w v + w is the family dened (v + w) (p) = (v + w)p = vp + wp ;

where

by

v + w : Rm → Sn0 : p 7→ vp + wp ,

i.e.

ii) the multiplication by scalars

· : C × s (Rm , Sn0 ) → s (Rm , Sn0 ) : (λ, v) 7→ λv where

λv

the sequel we shall 7) if

U

λv : Rm → Sn0 : p 7→ λvp m 0 denote s (R , Sn ) by X ;

is the family:

(λv) (p) = (λv)p = λvp

in

f ∈ L1loc (U, C), then Z hf | = hf |n : D(U, C) → C : g 7→ f gdµn

is an open subset of

Rn ,

i.e.

and

U

g ∈ L1loc (U, C) and f g ∈ L1 (U, C) Z hf |gin := hg|n (f ) := hf |n (g) := f gdµn ;

is the regular distribution generated by we put

f;

if

U

2

6= 8) S(h,ω) is the (h, ω) -Fourier-Schwartz transformation (where h, ω ∈ R = R\{0}) i.e. the operator S(h,ω) : Sn →Sn , such that, for all f ∈ Sn and a ∈ Rn , one has

 n Z  n 1 1 −iω(·|a) −iω(·|a) S(h,ω) (f )(a) = e fe dµn = (f ), h h Rn Rn .

where

(· | ·)

S(h,ω)

is an homeomorphism and, about its inverse, one has

is the standard scalar product on

− S(h,ω) (f )(a)

 =

|ω| h 2π

n Z

Moreover, we recall that

f eiω(·|a) dµn = S(2π/(|ω|h),−ω) (f )(a),

Rn

i.e.

 f (x) = With

R6= )

|ω| h 2π

F(h,ω)

n Z


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

Rn

we shall denote the

(h, ω)-Fourier

such that, for all

u ∈ Sn0

and for any

f ∈ Sn ,

h, ω ∈ F(h,ω) : Sn0 →Sn0 ,

transformation (where

on the space of tempered distributions, i.e. the operator one has

F(h,ω) (u)(f ) = u(S(h,ω) (f )), i.e. the transpose of

S(h,ω) ,

formally:

Moreover, we recall that

F(h,ω)

F(h,ω) = t (S(h,ω) ).

is an homeomorphism and that one has

− F(h,ω) = F(2π/(|ω|h),−ω) , and

F(h,ω) (u(α) ) = (ωi)α (·)α F(h,ω) (u)  α i α (F(h,ω) (u))(α) . F(h,ω) ((·) u) = ω 9) We shall use also the following denitions from [9]:

Denition 0.1 (family of tempered distributions of class S ). Let T ∈ s(Rm , Sn0 ) a family of distributions. One denes the family T family of m class S or S -family if, for each f ∈ Sn , the function T (f ) of R in C, dened m by T (f )(p) = Tp (f ), for each p ∈ R , belongs to the space Sn . The set of all these families is denoted by

S(Rm , Sn0 ) = S(m, Sn0 ) = S(m, n).

Denition 0.2 (of operator generated by an S - family of tempered distributions). Let T ∈ S(Rm , Sn0 ) be a family of tempered distributions of S.

T (or associated with T ) Tb : Sn → Sm dened by Tb(f )(p) = Tp (f ), for each f in Sn and m for each p in R , i.e. with the notations of the above denition, dened by b T (f ) = T (f ), for each f in Sn ;

class

One denes operator generated by the family

the operator

3

Denition 0.3 (family of tempered distributions of class SL). Let v ∈ S (Rm , Sn0 ) , a family of distributions. One denes the family v family of class SL or SL-family if one has v b ∈ L(Sn , Sm ). The family of such systems is denoted by

SL (Rm , Sn0 ) = SL(m, Sn0 ) = SL(m, n). 10) In [10] we have state and prove the following theorems:

Theorem 0.1 (of structure). vector space

S (Rm , Sn0 )

The set

is a subspace of the

(s(m, n), +, ·) .

Theorem 0.2 (of linear embedding).

The application

∧

(·) : S (Rm , Sn0 ) → Hom (Sn , S (Rm , C)) : v 7→ vb is an injective linear operator, thus one has

Theorem 0.3 (of structure). vector space

∧

(v + w) = vb + w b

The set

(S (Rm , Sn0 ) , +, ·) .

SL (Rm , Sn0 )

and

∧

(λv) = λb v.

is a subspace of the

Theorem 0.4 (of isomorphism). The function ∧

(·) : SL (Rm , Sn0 ) → L (Sn , S (Rm , C)) dened by

∧

(·) (v) = vb is

an isomorphism.

11) In [9] one gives the following denitions:

Denition 0.4 (linear superpositions of an SL-family). Let a ∈ S 0 (Rm , C) and

v ∈ SL (Rm , Sn0 ) .

v with respect to (the system a or ultralinear combination of v with respect to (the system a, or linear superposition of v with respect to (the system of

One denes generalized linear combination of of coecients) of coecients) coecients)

a,

the distribution

Z av := a ◦ vb : φ 7→ a (b v (φ)) ,

Rm i.e.

Z

av = t (b v )(a).

Rm Moreover, if

u ∈ Sn0

and there exists an

a ∈ S 0 (Rm , C)

such that

Z u=

av, Rm

u of

is said an

v

S 0 -linear

the distribution

v.

superposition of

Z

Z



v := Rm

Finally, we dene linear superposition

Rm

4

1(Rm ,C) v,



1(Rm ,C) is the regular m functional on R of value 1; where

distribution generated by the complex constant

Denition 0.5 (of S -ultralinear independence). denes

Let

v ∈ SL(Rm , Sn0 ).

One

v S -ultralinearly independent, if one has   Z u ∈ S 0 (Rm , C) ∧ uv = 0Sn0 ⇒ u = 0S 0 (Rm ,C) ; Rm

Denition 0.6 (of generalized linear span). One denes

v ∈ SL (Rm , Sn0 ) . S -ultralinear span of v , and it's denoted by Suspan (v), the   Z 0 0 m u ∈ Sn : ∃a ∈ S (R , C) : u = av ; Let

0

set

Rm

Denition 0.7 (system of S -ultragenerators). Let T dened system of

S -ultragenerators

for

V ⊆ Sn0

∈ SL (Rm , Sn0 ). T is Suspan (T ) = V ;

if and only if

Denition 0.8 (of S -ultrabasis). Let v ∈ SL(Rm , Sn0 ) and let V One denes

v S -ultrabasis of V

if it is

S

⊆ Sn0 .

-ultralinearly independent, and one

has

Suspan(v) = V ;

Denition 0.9 (the system of contravariant components).

Let v ∈ w ∈ Suspan(v). R 0 m av is The only tempered distribution a ∈ S (R , C) such that w = Rm denoted by [w|v] and is called the system of contravariant components of w with respect to v or the system of coordinates of w in v .

SL(Rm , Sn0 )

1.

be an

S

-ultralinearly independent family and

S 0 -operators

Denition 1.1 (operator of class S 0 ). Let A ∈ Hom(Sn , Sm ). One denes

A

operator of class S 0 or S 0 -operator if the functional δp ◦ A is a tempered

distribution for all p ∈ Rm , i.e.

δp ◦ A ∈ Sn0 ,

for each p ∈ Rm . 

Example 1.1 Each A ∈ L(Sn , Sm ) is an operator of class S 0 .

4

Notation 1.1 The set of all S 0 -operators is denoted by S 0 Hom(Sn , Sm ). Theorem 1.1 (of structure). The set Hom(Sn , Sm ).

5



S 0 Hom(Sn , Sm ) is a subspace of

Proof.

Let

λ∈C

and

A, B ∈ S 0 Hom(Sn , Sm ).

(δp ◦ (A + λB)) (φ)

For each

p ∈ Rm ,

one has

= δp ((A + λB) (φ)) = = δp (A (φ) + λB(φ)) = = δp (A (φ)) + δp (λB(φ)) = =

(δp ◦ A)(φ) + λδp (B(φ)) =

=

[δp ◦ A + λ (δp ◦ B)] (φ) ,

so

δp ◦ (A + λB) = δp ◦ A + λ (δp ◦ B) and then

δp ◦ (A + λB) ∈ Sn0 . 

Theorem 1.2 (of isomorphism). Let

tor of class S , moreover the application 0

v ∈ S(m, Sn0 ). Then, vb is an opera-

∧

(·) : S(m, Sn0 ) → S 0 Hom(Sn , Sm ) : v → vb

is an isomorphism. Proof.

Let

p ∈ Rm ,

one has

(δp ◦ vb) (φ)

=

δp (b v (φ)) =

= vb(φ)(p) = = vp (φ). So, one has

δp ◦ vb = vp ∈ Sn0 , and hence

vb is

an operator of class

S 0.

Now, we already know that

∧

(·) : S(m, Sn0 ) → Hom(Sn , Sm ) is an injective homomorphism (see the preliminaries) hence we have to prove that the application

∧

(·) : S(m, Sn0 ) → S 0 Hom(Sn , Sm ) is surjective. Let

A ∈ S 0 Hom(Sn , Sm )

and let

v : Rm → Sn0 : p 7→ δp ◦ A, be a family of distributions. For each

v (φ) (p)

φ ∈ Sn

and for each

=

vp (φ) =

=

(δp ◦ A)(φ) =

=

δp (A (φ)) =

=

A (φ) (p) 6

p ∈ Rm ,

one has

thus

v (φ) = A (φ) and hence, in particular, we deduce that

v (φ) ∈ Sm , and so

v ∈ S(Rm , Sn0 ). Moreover, for any

φ ∈ Sn ,

one has

vb(φ) = v(φ) = A(φ) and thus

vb = A. Q.E.D. 

2.

The family generated by an

Denition 2.1 (of family generated by an

S Hom(Sn , Sm ). One denes 0

S 0 -operator

family generated by

S 0 -operator). Let A ∈ A the family

A∨ = (δp ◦ A)p∈Rm . 

Example 2.1 Let (·)S

n

be the identic operator on

[(·)Sn ]∨

Let

S(a,b)

the

(a, b)-Fourier

p ∈ Rn ,

(δp ◦ (·)Sn )p∈Rn =

=

(δp )p∈Rn .


∨

= δp ◦ S(a,b)

= = =

S(a,b)


∨


p∈Rn

,

one has

δp ◦ S(a,b)

and thus

one has

Schwartz transformation, then, one has

S(a,b) now, for each

=

Sn ,

is the

(a, b)-Fourier

t


S(a,b) (δp ) =

F(a,b) (δp ) =  n 1 e−ib(p|·) a family.

Theorem 2.1 (of isomorphism). Let

family A is an S -family and the application ∨

4 A ∈ S 0 Hom(Sn , Sm ). Then, the

∨

(·) : S 0 Hom(Sn , Sm ) → S(Rm , Sn0 )

is the inverse of the isomorphism (·)∧ of the theorem 1.2. 7

Proof.

For all

p ∈ Rm ,

one has

(A∨ )p = δp ◦ A, so, for each

φ ∈ Sn ,

one has

(A∨ ) (φ) (p)

=

(A∨ )p (φ) =

=

(δp ◦ A)(φ) =

= δp (A(φ)) = = A (φ) (p) and hence

(A∨ ) (φ) = A (φ) ∈ Sm , thus

A∨

is an

S -family.

Moreover, let

A ∈ S 0 Hom(Sn , Sm ), ∧

(A∨ ) (φ) (p)

and so

one has

=

(A∨ )p (φ) =

=

(δp ◦ A) (φ) =

=

δp (A (φ)) =

=

A (φ) (p)

∧

(A∨ ) = A. Let

v ∈ S(m, n)

one has

∨

((b v ) )p = δp ◦ vb = vp and thus

(b v )∨ = v;

in fact, the last equality holds because

(δp ◦ vb) (φ)

=

δp (b v (φ)) =

= vb (φ) (p) = = vp (φ) , and this completes the proof.

3.



Generated families and S -ultrabases

Theorem 3.1 (fundamental theorem on the coordinates in a generated family). Let A ∈ L(Sn , Sn ) be a bijective operator. Then, A∨ is an

ultrabasis of Sn0 . Moreover, for each u ∈ Sn0 , one has Z u=

(u ◦ A− )A∨ ,

Rn

8

and so

[u | A∨ ] = u ◦ A− .

Proof.

Let

A−

the inverse of

A,

one has

A− ∈ L(Sn , Sn ), in fact, because

Sn is a Frechét space, it follows by the Banach's inverse operator u ∈ Sn0 , one has

theorem. Now, for any

u ◦ (·)Sn =

u = =

u ◦ (A− ◦ A) =

=

(u ◦ A− ) ◦ A =

=

(u ◦ A− ) ◦ (A∨ )∧ = Z (u ◦ A− )A∨ ,

=

Rn and thus

Suspan(A∨ ) = Sn0 , so

A∨

Sn0 . ∨ that A

ultragenerates

Now we prove Let

a ∈ Sn0

is ultralinearly independent.

be such that

Z Rn

aA∨ = 0Sn0 ,

recalling the denition of ultralinear combination, one has

0Sn0

= a ◦ (A∨ )∧ = = a ◦ A,

and hence

0Sn0

=

0Sn0 ◦ A− =

=

(a ◦ A) ◦ A− =

= a ◦ (A ◦ A− ) = = a ◦ (·)Sn = a, so

a = 0Sn0

and

A∨

is also ultralinearly independent.

Example 3.1 The operators (·)S thus the families

[(·)Sn ]∨

n

and

S(a,b) ∨

and

S(a,b)

Q.E.D.

are two bijective operators and

are two ultrabases of

has

[u| [(·)Sn ]∨ ]

=

u ◦ [(·)Sn ]− =

=

u ◦ (·)Sn =

=

u, 9



Sn0 .

Moreover, one

and

[u| S(a,b) ∨ ]

= u ◦ S(a,b)


−

=

−

= F(a,b) (u). 4 In such way, the Fourier ultraexpansion theorem (see [Ca]) is a particular case of the above theorem. At this point, we recall that if

A ∈ L(Sn , Sm ),

the

operator

t is called the transpose of

0 A : Sm → Sn0 : a 7→ a ◦ A

A.

Corollary 3.1 Let

ultrabasis of

Sn0

A ∈ L(Sn , Sn ) be a bijective operator. Then, A∨ is an and moreover, for each u ∈ Sn0 one has Z t u= (A− )(u)A∨ , Rn

where t (A− ) is the transpose operator of A, and hence one has [· | A∨ ] = t (A− ).

Example 3.2 One has [·| [(·)Sn ]∨ ]

=

t

=

t

= and

[·| S(a,b) ∨ ] =

4.

t





 − (·)Sn =
(·)Sn =

(·)Sn0 ,

 − − S(a,b) = F(a,b) . 4

Bases and transpose operators

The theorem 3.1 we have another corollary, to state it we give a denition due to D.Carfì:

Denition 4.1 (superposition of a family with respect to an operator). Let V ⊆ Sn0 be a subspace of Sn0 , A ∈ Hom(V, Sm0 ) and v ∈ SL Rm , S 0 Rk , C

be a family of distributions. One denes A, the operator Z Av : V → Rm

Sk0

superposition of v with respect to Z

: u 7→

A(u)v.  Rm

10





Example 4.1 (on the position operator). be the position operator and let

Z

f

be the

 Xf

Let

(a, b)-Fourier

X : S10 → S10 : u 7→ (·)u family, then one has

Z (u)

=

X(u)f =

R

ZR =

[(·)u]f =   t b f ((·)u) =
t S(a,b) ((·)u) = R

= =

F(a,b) ((·)u) =  1
0 i F(a,b) (u) = b
i P F(a,b) (u) = b −i~
−1 P ◦ F(a,b) (u) b~

= = = = and hence

Z Xf = R


−1 P ◦ F (a,b) , b~

where

P : S10 → S10 : u 7→ −i~u0 is the momentum operator of a quantum particle.

4

Example 4.2 (on the momentum operator). Z

 Pf

(u)

=

R

P (u)f = ZR

=

(−i~u0 )f = R   t b f (−i~u0 ) =
t S(a,b) (−i~u0 ) =

=

F(a,b) (−i~u0 ) =

=

−i~F(a,b) (u0 ) =

=

−i~(ib)1 (·)1 F(a,b) (u) =

=

~b(·)F(a,b) (u),

= =

and hence

One has

Z

Z


P f = b~ X ◦ F(a,b) . 4

R

Example 4.3 (on the kinetic energy operator). T : S10 → S10 : u 7→ 11

~2 00 u 2m

Let

be the kinetic energy operator of a nonrelativistic quantum particle, one has

Z

 Tf

Z (u)

=

T (u)f =

R

R

~2 00 u f= R 2m    ~2  t b f u00 = 2m  2  ~ 00 F(a,b) u = 2m ~2 (ib)2 (·)2 F(a,b) (u) = 2m
~2 b2 − X 2 ◦ F(a,b) , 2m Z

= = = = = and hence

Z Tf = − R


~ 2 b2 X 2 ◦ F(a,b) . 4 2m

The corollary of theorem 3.1 is the following.

Corollary 4.1 (resolution of the identity). Let A ∈ L(Sn , Sn ) be a bijective operator. Then, A∨ is an ultrabasis of Sn0 and moreover one has Z (·)Sn0 =

t

(A− )A∨ .

Rn

More generally, we have the following theorem.

Theorem 4.1 (resolution of a transpose operator). Let A, B ∈ L(Sn , Sn )

be two bijective operators. Then,

A∨ · B ∨

is an ultrabasis and t

Z B=

t

(A− )(A∨ · B ∨ ).

Rn

Proof.

The family

A∨ · B ∨

is a basis because one has (see the remark below)

A∨ · B ∨ = (A ◦ B)∨

12

and

A◦B

u ∈ Sn0 , one  t − ∨ t (B) (A )(u)A = Rn Z  t (A− )(u)A∨ ◦ B =

is a bijective operator. Now, for each

t

has

Z

(B)(u)

= = =

Rn −

t

[ (A )(u) ◦ A] ◦ B = t

=

(A− )(u) ◦ (A ◦ B) =

Z =

t

(A− )(u)(A ◦ B)∨ =

t

(A− )(u)(A∨ · B ∨ ). 

Rn

Z = Rn

Remark 4.1 If A, B ∈ L(Sn , Sn ) and p ∈ Rn , one has (A ◦ B)∨ (p)

= δp ◦ (A ◦ B) = (δp ◦ A) ◦ B = Z = (δp ◦ A)B ∨ = Rn Z = (A)∨ (p)B ∨ =

=

Rn

(A∨ · B ∨ )p . N

=

After a denition of D.Carfì, we give another theorem.

0 Sm

Denition 4.2 (image of a family under an operator). Let be an operator and v : R → k

Sn0

A : Sn0 →

be a family of distributions. The family

0 A(v) : Rk → Sm : p → A(vp )

is called

image of v under A.



Theorem 4.2 (on the image under a transpose operator). Let

0 ). Then, L(Sn , Sm ) and v ∈ SL(Rk , Sm t

Proof.

For each

p ∈ Rk ,

B(v) = v · B ∨ .

one has

(v · B ∨ )p

Z

vp B ∨ =

= R

= vp ◦ (B ∨ )∧ = = vp ◦ B = =

t

B (vp ) =

=

t

B(v)(p)

13

B∈

and hence

v · B ∨ = t B(v). 

Theorem 4.3 (ultralinearity of a transpose operator). Let

0 L(Sn , Sm ) and v ∈ SL(Rk , Sm ). Then, for each a ∈ Sk0 one has Z  Z t B av = a t B(v). Rk

Proof.

B ∈

Rk

One has

t

Z



B

av

Z

 av ◦ B =

=

Rk

Rk

(a ◦ vb) ◦ B =

=

= a ◦ (b v ◦ B) = Z = a(b v ◦ B)∨ = Rk Z = a(v · B ∨ ) = Rk Z = a t B(v).  Rk

Remark 4.1

The above theorem furnishes a property very similar to the

following classic property of a vector space:

if A ∈ Hom(V, W ) then

A

k X

! ai vi

=

i=1

k X

ai A(vi );

i=1

so we can consider the above property as an ultralinearity of the transpose of an operator

B ∈ L(Sn , Sm ). N

We conclude this section with a characterization of the ultrabases by the transpose operators.

Theorem 4.4 (characterization of the ultrabases). Let A ∈ Hom(Sn , Sm ). Then, the following assertions hold 1) A∨ is ultralinearly independent if and only if t A is injective; 2) A∨ ultragenerates Sn0 if and only if t A is surjective; 3) A∨ is an ultrabasis if and only if t A is bijective.

14

Proof.

1)

A∨

is ultralinearly independent if and only if

Z Rm implies

0 a = 0Sm

0 a ∈ Sm

and

aA∨ = 0Sn0

but, because

Z

aA∨ = a ◦ A = t A(a),

Rm this is equivalent to the statement:

t

A(a) = 0Sn0

implies

0 , a = 0Sm

and then

ker t A = {0Sn0 }. 2)

A∨

ultragenerates

Sn0

if and only if, for each

such that

Z u=

u ∈ Sn0

there exist an

0 a ∈ Sm

aA∨ = t A(a)

Rm but this is the surjectivity of

t

A.

3) It follows directly from the above two assertions.



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David Carfì

Clara Germanà

Via Canova, 32

Institute of Mathematics

98121 Messina,

Faculty of Economy

Sicily, Italy

University of Messina

david.car@cys.it

Via Dei Verdi, 75

[email protected]

98122 Messina, Sicily, Italy

dcar@dipmat.unime.it

[email protected]

17