Injective tensor product

Summary

Let $X$ and $Y$ be Banach spaces. Their injective tensor product is the completion of the algebraic tensor product $X \otimes Y$ for the smallest reasonable norm on it. This norm sees a tensor only through its values on pairs of functionals of norm at most $1$. It describes spaces of continuous vector-valued functions: $C(K)$ with the injective tensor product by $X$ is the space of continuous functions from $K$ to $X$. With the projective tensor product it is one of the two extreme tensor norms of Grothendieck's theory.