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

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up vote 6 down vote accepted

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
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Yeah, that's so much clearer than my intuition! Thanks. –  Jonas Meyer Jan 5 '10 at 4:46
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