Usual norm & metric on a (finite) product of normed spaces

Question

I'm studying from Garling's Course in Mathematical Analysis, and in many questions they don't specify what norm I should use to prove statemens like continuity in products of normed spaces. The norms of the spaces themselves are arbitrary but clearly if something about the product is asked, the corresponding norm should relate to the norms of the spaces themselves. It would be something in the manner of the following:

$(E_1,\|\cdot\|_1)$, $(E_2, \|\cdot\|_2)$ and $(E_3,\|\cdot\|_3)$ be normed spaces, and $f:E_1\times E_2\to E_3$ be a mapping (with some additional details). Prove that $f$ is continuous.

An actual example: Let $(E,\|\cdot\|)$ be a real normed space. Show that the mapping $f:E\times E\to E$ given by $f(x,y)= x+y$ is uniformly continuous - What norm should be considered on $E\times E$ ?

In this type of question, I usually end up using the maximum norm and metric and it tends to work out. I know the $p$-metric is equivalent to all the metrics it induces like the taxicab metric, but also the metric induced by the maximum norm.

Is there another metric or norm that is meant in these kinds of questions, that I'm just overlooking, or is my approach correct?

Answer 1

So topology is a Tichonoff product topology, I guess. Correct me if I'm wrong. Given any norm on 2 dimensional space $\mathbb{R}^{2}\ni(x,y)\mapsto |(x,y)|\in \mathbb{R}_{+}$, the norm on product given by $E_{1}\times E_{2}\ni(x,y)\mapsto |(\|x\|_{1},\|y\|_{2})|\in\mathbb{R}_{+}$ induces product topology. And all norms of that form are equivalent (they are equivalent to for example p-norm). But I dont know if it's true that any norm on a finite product induces same topology. I bet it's not true, since such a product can be infinitely dimensional, so to get counterexample, you have to find an infinitely dimensional space and 'split it' into a product of two spaces, which is probably doable.