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## Homework Statement

I'm reading the proof of a theorem and the author claims w/o justification that a weakly lower semi-continuous function (w.l.s.c.) f:C-->R attains its min on the convex weakly compact subset C of a normed space E.

At first I though I saw why: Let a be the inf of f on C and x_n be a sequence in C such that f(x_n) --> a. Since C is weakly compact,

**we can find a weakly convergent subsequence**x_n_k-->x, and because f is w.l.s.c., we will have f(x)<=a, thus f(x)=a.

But what reason do we have to believe that C is

**weakly sequentially compact**, so that the bold part above is justified??

(By "weakly" I mean "under the weak topology [tex]\sigma(C,C^*)[/tex]".)

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