Projective tensor product

Summary

Let $X$ and $Y$ be Banach spaces. Their projective tensor product is the completion of the algebraic tensor product $X \otimes Y$ for the largest reasonable norm on it. It turns bounded bilinear maps on $X \times Y$ into bounded linear maps, with the same norm. Its dual is the space of bounded linear maps from $X$ to the dual of $Y$. With the injective tensor product it is one of the two extreme ways to make $X \otimes Y$ a Banach space, as studied by Grothendieck.