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
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
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