subsequence of bounded sequence in a finite dimension banach space

Question

I have doubt: if $E$ is a finite dimensional Banach space and I take a bounded sequence in its dual $E^\star$, $\{f_n\}$ (with $||f_n||=1$), could I deduce that there exists a convergent subsequence $\{f_{n_k}\}$ ?

Answer 1

Yes, because a closed ball in a finite dimensional normed space is compact. A way to see this is to remark that every norm in a finite dimensional normed space are equivalent. This implies that a closed ball in $(E,\|\|)$ is included in a closed Euclidean ball. You have $\|x-c\|_e\leq h\|x-c\|$ where $\|\|_e$ is the Euclidean norm. Thus $B(c,r)\subset B_e(c,hr)$. Since a closed ball in the Euclidean space is compact, you deduce that closed ball in $(E,\|\|)$ are compact, henceforth, you can extract a converging sequence from anya bounded sequence in $(E,\|\|)$.