While trying to generalise the well-known result "continuous partials imply differentiability" to the case of infinite-dimensional spaces, the following question popped up in my work:
Let $V$ be a normed linear space over $\mathbb K$ with a basis $\mathcal B$. Is the induced isomorphism $T\colon V\to \mathbb K^{[\mathcal B]}$ continuous (with respect to any given norm on $\mathbb K^{[\mathcal B]}$)?
It's clear that for any norm $\lVert\cdot\rVert$ on $\mathbb K^{[\mathcal B]}$, there exists a norm on $V$, namely, $v\mapsto \lVert Tv\rVert$, which makes $T$ continuous: Just note that $T$ is a linear map which is bounded.
Thus, due to equivalence of norms on finite-dimensional vector spaces, if $\dim V < \infty$, then the answer to the posed question is affirmative.
However, I am unable to comment anything for a general norm on an infinite-dimensional $V$. Any ideas?