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Mathematical theorem in convex analysis
A function that is not
lower semi-continuous . By the Fenchel-Moreau theorem, this function is not equal to its
biconjugate .
In
convex analysis , the FenchelâMoreau theorem (named after
Werner Fenchel and
Jean Jacques Moreau ) or Fenchel biconjugation theorem (or just biconjugation theorem ) is a
theorem which gives
necessary and sufficient conditions for a function to be equal to its
biconjugate . This is in contrast to the general property that for any function
f
∗
∗
≤
f
{\displaystyle f^{**}\leq f}
.
[1]
[2] This can be seen as a generalization of the
bipolar theorem .
[1] It is used in
duality theory to prove
strong duality (via the
perturbation function ).
Statement
Let
(
X
,
τ
)
{\displaystyle (X,\tau )}
be a
Hausdorff
locally convex space , for any
extended real valued function
f
:
X
→
R
∪
{
±
∞
}
{\displaystyle f:X\to \mathbb {R} \cup \{\pm \infty \}}
it follows that
f
=
f
∗
∗
{\displaystyle f=f^{**}}
if and only if one of the following is true
f
{\displaystyle f}
is a
proper ,
lower semi-continuous , and
convex function ,
f
≡
+
∞
{\displaystyle f\equiv +\infty }
, or
f
≡
−
∞
{\displaystyle f\equiv -\infty }
.
[1]
[3]
[4]
References
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