While studying some notes on normed vector spaces, I have come upon the proof that addition $+:V \times V\to V$ of vectors in a normed vector space $V$ is a continuous operation.
The proof of this fact is quite easy except of (in my opinion) one step: the choice of the norm on $V \times V$. The proof has been done for a norm defined as $\|(v_1,v_2)\| = \|v_1\| + \|v_2\|$ and a comment has been made that certain other norms (e.g. $\|(v_1,v_2)\| = \max\{\|v_1\|,\|v_2\|\}$) can be used as well, since they generate the same topology (the product topology, I suppose).
However, I struggle with the question what is the precise set of all norms that can be used in this proof. I suppose that the theorem has to be interpreted in the way that $+$ is continuous with respect to the product topology. Thus, my question can be restated as follows: given a norm $\|\cdot\|$ on $V$ generating a topology $\tau$, which norms can be used on $V \times V$ to generate the product topology with respect to $\tau$?
I do not find this question to be straightforward, since in infinite dimensional spaces, norms need not be equivalent.
Thank you in advance.
Best Answer
Any Norm $p$ on $\mathbb R^2$ (you know that the are all equivalent) gives you a norm $(v,w)\mapsto p(\|v\|,\|w\|)$ on $V\times V$ which is equivalent to $\|v\| +\|w\|$ and induces the product topology. Your question could be whether all equivalent norms are of that form.