Let $X$ be an infinite dimensional vector space over a field $\mathbb{K}$. Suppose that $(X,\|\cdot\|)$ is a complete normed vector space, in the sense that any Cauchy sequence is convergent. Suppose that the closed unit ball of $X$ is compact in the strong topology.

** Question 1.**
Is $X$ necessarily isomorphic to some finite dimensional Banach space ?

** Question 2.** If the answer for the question 1 is ** no **, can we always find in $\mathcal{L}(X,X)$ an unbounded linear operator ?

Different from this question Is there an infinite-dimensional Banach space with a compact unit ball? I would like to assume the axiom of choice.

**Comment. **
The example I have in mind is $(\mathbb{R}^n,\|\cdot\|_{2})$ as $\mathbb{Q}$ vector space.
Clearly
$\text{dim}_{\mathbb{Q}}\ \mathbb{R}^n=\infty$ and the unit ball is compact and this space is complete with respect to the standard Euclidean norm $\|\cdot\|_2$, but in this example both questions 1 and 2 are trivial.

toassume the" $\endgroup$ – user5810 Oct 13 '11 at 6:495more comments