There is a vector space $X$ with two norms $\|\cdot\|_1, \|\cdot\|_2$ such that $(X,\|\cdot\|_1)$ is separable and $(X,\|\cdot\|_2)$ is not?
In both cases, we will consider the topology induced by norm.
2026-03-26 19:03:52.1774551832
Separable Space
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Hint: Let $X$ be the vector space of bounded real sequences. Consider, for $x= (x_n) \in X,$
$$\|x\|_1 = \sum_{n=1}^{\infty}\frac{|x_n|}{2^n}, \,\, \|x\|_2 = \sup_{n\in \mathbb N}|x_n|.$$