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Riesz's lemma:

Let $Y$ and $Z$ be subspaces of a normed space $X$ and suppose $Y\neq Z$. Then for every $\theta\in (0,1)$ there is $z\in Z$ such that $\|z\|=1$ and $\|z-y\|\geq\theta$ for all $y$.

Using Riesz lemma prove that

A normed space is locally compact iff it is finite dimensional.

How to proceed ? Should I use the fact that

The unit ball is compact in a normed linear space iff the space is finite-dimensional.

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  • Also relevant: http://math.stackexchange.com/questions/207839/a-finite-dimensional-normed-space – Alex M. Sep 25 '16 at 11:43
  • It seems like the last statement is what you're trying to prove. Heine-Borel deals with the finite dimensional case. For the infinite dimensional space, use riesz' lemma to construct a sequence behaving badly. – SamM Sep 25 '16 at 11:45

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