TITLE: Is vector addition continuous under arbitrary norms
QUESTION [6 upvotes]: Suppose $X$ is a normed space, and $X\times X$ is the vector space with vector addition and such defined component-wise, equipped with some norm. Then is $(x, y) \mapsto x + y$ continuous?
I know that if the norm is just $(x, y) \mapsto |x| + |y|$ then this is satisfied, but what if theres some other norm?

REPLY [2 votes]: If $X$ is a finite-dimensional vector space over $\Bbb{R}$ then any two norms on $X$ determine the same topology and vector addition will be continuous with respect to that that topology.
If $X$ is infinite-dimensional then different norms on $X$ can give different topologies on $X$ and you can't expect vector addition to be continuous if you choose an arbitrary norm on $X \times X$.