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

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.

• I'm not sure why you've added the definition. Your question asks for something which is not really related to the definition. – Aryaman Maithani Jun 6 at 16:01

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.
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 $$2^{\dim X}$$.