The problem is
Prove that a Banach space is finite-dimensional if and only if each of its subspace is closed.
The 'only if' part is quite easy. Since any finite dimensional space is complete, the subspace must also be complete and thus closed.
I don't know how to deal with the 'if' part, though I found this. I don't understand the idea of it. And the treatment is beyond my scope.
Are there any Hints or ideas? Thank you in advance.
Best Answer
Let $\{e_n\}$ be a countable linearly independent set in $X.$ Let $X_0$ denote the linear span of $\{e_n\}$ and $X_1$ denote the closure of $X_0.$ Then $X_1$ is an infinite dimensional Banach space. As its Hamel basis is uncountable (the fact which follows from the Baire category theorem), then $X_0\subsetneq X_1$ and $\bar X_0=X_1.$ Therefore $X_0$ is not closed.