# What are the products in the category of normed vector spaces with linear contractions?

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

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## 1 Answer

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.)

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One way to arrive at this answer is to note that Hom(R, X) is the set of points in the closed unit ball in X, so the unit ball in X x Y must be the product of the unit balls in X and Y. –  Reid Barton Jan 5 '10 at 4:40
Yeah, that's so much clearer than my intuition! Thanks. –  Jonas Meyer Jan 5 '10 at 4:46