Does there exist an uncountable dimensional real vector space $X$ such that $(X,\|\cdot\|)$ is a Banach space for any norm $\|\cdot\|$ on it?

Question

Does there exist an uncountable dimensional real vector space $X$ such that $(X,\|\cdot\|)$ is a complete space for any norm $\|\cdot\|$ on it ?

Answer 1

No. On every infinite-dimensional real (or complex) vector space, there are comparable but not equivalent norms, i.e. there are norms $\lVert\,\cdot\,\rVert_1$ and $\lVert\,\cdot\,\rVert_2$ with

$$\lVert x\rVert_1 \leqslant \lVert x\rVert_2$$

for all $x\in X$, but there is no $C \in (0,+\infty)$ with

$$\lVert x\rVert_2 \leqslant C\cdot \lVert x\rVert_1$$

for all $x\in X$. By the open mapping theorem, at most one of $\lVert\,\cdot\,\rVert_1$ and $\lVert\,\cdot\,\rVert_2$ can make $X$ into a Banach space.

To see the existence of such norms, consider a Hamel basis $\{ e_{\alpha} : \alpha \in A\}$ of $X$ and take two functions $f_1,\,f_2 \colon A \to (0,+\infty)$ with $f_1(\alpha) \leqslant f_2(\alpha)$ for all $\alpha\in A$ and $\frac{f_2(\alpha)}{f_1(\alpha)}$ unbounded. Then define

$$\Biggl\lVert \sum_{\alpha \in A} c_{\alpha}\cdot e_{\alpha}\Biggr\rVert_i = \max \{ f_i(\alpha)\cdot \lvert c_\alpha\rvert : \alpha \in A\}.$$