Find all vector spaces $X$ such that every two norms are equal

Question

Find all vector spaces $X$ such that for any two norms $\|\cdot\|_{a},\|\cdot\|_{b}$ there exist positive constant $ C$ such that for all $x \in \mathbb{V}$ we have $\|x\|_{a} =C\|x\|_{b}$.

Definition. Let $\mathrm{V}$ be a vector space over the real or complex numbers. Let $\|\cdot\|_{a},\|\cdot\|_{b}$ be norms. We say that $\|\cdot\|_{a},\|\cdot\|_{b}$ are equivalent if there exist positive constants $c, C$ such that for all $x \in \mathbb{V},$

$$c\|x\|_{a} \leq\|x\|_{b} \leq C\|x\|_{a}$$

I think this is true for every $1$-dimensional vector space.

Answer 1

If $\dim X=1$ then this property follows immediately from $\|cv\|=|c|\|v\|$.

If $\dim X\ge 2$, write $X=k\oplus Y$ and note that for any norm $\|\cdot\|$ on $k\oplus Y$, we can define $\|(c,y)\|':=\|(2c,y)\|$ to obtain something equivalent but non-equal.

Answer 2

If $X$ is finite-dimensional, then any all norms on $X$ are equivalent.

If $X$ if infinite-dimensional, then $X$ always has at least two inequivalent norms. In fact, it was shown by M. Kwon, that the number of equivalent norms on $X$ is exactly $2^{\dim X}$.

On a related note, if $X$ is a real vector space of (infinite) dimension $\kappa$, and that there exists at least one complete norm on $X$. Then there are $2^\kappa$ inequivalent complete norms on $X$.