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 XiiinI is a family of normed vector spaces with all norms ambiguously denoted |cdot|, then the product is X=aiinprodXi:sup|ai|<infty, and for aiinX, |ai|=sup|ai|.
(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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