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My friends,

I have a question about Fenchel's duality.

Background: According to Wiki, in Fenchel duality, we have the following theorem:

Let $X$ and $Y$ be Banach spaces, $f: X \rightarrow \mathbb{R} \cup \{+ \infty\}$ and $g: Y \rightarrow \mathbb{R} \cup \{+ \infty\}$ be convex functions and $A: X \rightarrow Y$ be a bounded linear map. Then, the Fenchel problems are: $$ p^* = \inf_{x \in X} \{ f(x) + g(Ax) \} \\ d^* = \sup_{y^* \in Y^*} \{ -f^*(A^*y^*) - g^*(-y^*) \} $$ and $p^* = d^*$ if strong duality holds.

My question is:

In the above duality, if $X = [0,1]^n$ (in other words, each $x \in X$ is a $n$-dimensional vector with each dimension's value as 0~1), what will be the space Y*?

Particularly, will this $Y^*$ space be non-negative (in other words, should each $y^*$ be some non-negative vector)?

Thanks a lot for your attention and very appreciate it if you can kindly help me with this!


share|cite|improve this question
$[0, 1]^n$ is not a Banach space. – Qiaochu Yuan Apr 23 '13 at 17:43
@Qiaochu: thanks for correcting the error. Let's say, if $X \in \mathbb{R}^n$, but in practice assume $x$ is some $n$-dimensional vector with each dimension's value as 0~1, then is there any way to know whether those $y^*$ should be non-negative? – Vincent Apr 24 '13 at 7:02

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