A finite dimensional normed space 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.

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.