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

