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I would like to find an elementary proof of the following theorem

Let $E$ be a normed space. Then the following statements are equivalent:

(a) E is finite dimensional.

(b) Every linear functional on $E$ is continuous.

(c) Every linear subspace of $E$ is closed.

I would like to prove $(a)\Leftrightarrow (b)$ and $(b)\Leftrightarrow (c)$

Thank you for all helping.

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It depends on what you mean by "elementary"; you can't do it without the axiom of choice. If I recall correctly, it's consistent with ZF+DC that (b) is true in every separable Banach space. –  Nate Eldredge Oct 18 '12 at 4:21
    
@Nate Eldredge : Dear Sir. Elementary here means direct proof. –  blindman Oct 18 '12 at 4:33
    
See here. –  Mhenni Benghorbal Oct 18 '12 at 4:44

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