In the category of normed vector spaces in which the morphisms are linear contractions, what do products look like?



What are the products in the category of normed vector spaces with linear contractions?
Two words: sup norm.


I.e., the product of a family is the uniformly bounded subset of the cartesian product of the family, and the norm is the smallest uniform bound.


Explicitly, if $\{X\_i\}\_{i\in I}$ is a family of normed vector spaces with all norms ambiguously denoted $\|\cdot\|$, then the product is $X=\{\{a\_i\}\in\prod X\_i:\sup\|a\_i\|<\infty\}$, and for $\{a\_i\}\in X$, $\|\{a\_i\}\|=\sup\|a\_i\|$.


(My intuition came from products of C\*-algebras, where the $\*$-homomorphisms are automatically contractive and products are defined in this way. So I had a good guess and it is easy to check that it works.)


