Contemporary Mathematics

Volume 4315, 2007

Young-Fenchel transformation and some new characteristics

of Banach spaces

Ya. I. Alber

ABSTRACT.

By making use of the Young-Fenchel transformation for conju-

gate functions,

we

derive new characteristics of Banach spaces such as the

/-generalized projection operators and decompositions of potential operators,

the equivalence theorems for variational inequalities and Pythagorean type

theorems. We study their properties and present some applications.

1.

Introduction

Assume that

B

is a real reflexive Banach space,

B*

is its dual space,

(}B

and

(}B.

are origins of Band

B*,

n

is a convex closed subset of

B.

Let

J : B--+ B*

and

J* : B*

--+

B

be normalized duality mappings in

B

and B*, respectively,

Pn : B

--+

n

be a metric projection operator, lin : B

--+

n

be a generalized

projection operator onto

n

[4].

Let x be an arbitrary element of B and

1/J

be an

arbitrary element of

B*.

In

[6]

we have established the following decompositions of

x

and

1/J:

If K

C

B

is a closed convex cone with the vertex

(}B

and K°

C

B*

is its

polar cone with the vertex

(}B.

(see Definitions 3.5 and 3.7), then

with

and

{1.1)

with

X= PKX

+

J*IIKoJx

(PKo't/J,

ITKJ*'IjJ) =

0,

where

(/,

x) denotes the dual product of elements

/

E

B*

and

x

E

B.

It is obvious that if

f :

B

--+

R

is a differentiable functional and

f' :

B

--+

B*

is its finite Gateaux derivative, then

/'(x) = PKof'(x)

+

JIIKJ* f'(x).

1991 Mathematics Subject Classification. Primary 46B25, 47H30, 47J20, 49M27; Secondary

47H50, 49J40, 90046.

Key words and phrases. Banach spaces, Young-Fenchel transformation, conjugate functions,

generalized projections,

potential opetators, variational inequalities, Pythagorean theorems, de-

composition theorem.

@2007 American Mathematical Society

1

http://dx.doi.org/10.1090/conm/435/08362