---
tag: 685ZV73JJ8
kind: concept
name: Projective tensor product
msc: [46B28, 46M05]
specializes: [0BF2DE2SYK]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_tensor_product
same_as:
  wikidata: Q89672221
  wikipedia: Projective tensor product
---

## Summary

Let $X$ and $Y$ be Banach spaces.
Their projective tensor product is the completion of the algebraic [[tensor product|0BF2DE2SYK]] $X \otimes Y$ 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|RQXC5D3M1Y]] it is one of the two extreme ways to make $X \otimes Y$ a Banach space, as studied by Grothendieck.

## Explanations

### L1 human

A tensor $u$ in $X \otimes Y$ can be written in many ways as a sum of simple products $x_i \otimes y_i$.
Each way has a cost, the sum of the products of the lengths $\norm{x_i} \, \norm{y_i}$.
The projective norm of $u$ is the cheapest cost over all ways to write it.
For example, in $\ell^1 \otimes \ell^1$ this norm turns a tensor into a double sequence $(a_{mn})$ with norm $\sum_{m,n} \abs{a_{mn}}$.