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I would like to find a short proof for the following theorems:

Theorem 1. A normed space is finite dimensional iff all of its linear functional is continuous.

Theorem 2. A normed space is finite dimensional iff its unit ball is compact.

Thank you in advance.

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    Ok. For the second theorem take a look on the page 160 of "Brezis - Functional Analysis, Sobolev Spaces and PDE". For the first one is like Davide Giraudo said.2012-10-05

1 Answers 1

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The direction you asked for in the comments:

Let $X$ be an infinite dimensional normed space.

  1. Pick a countable independent collection $(e_n)_{n\in\mathbb{N}}$, pick $(y_i)_i$ such that $(e_n,y_i)_{n,i}$ is a basis. Let $f$ be the functional determined by $f(e_n)=n\|e_n\|$ and $f(y_i)=0$. Then $f$ is unbounded.

  2. By Riesz's lemma one easily constructs a sequence of independent vectors in the unit ball without a converging subsequence.