---
tag: EY0X2NKA3S
kind: concept
name: Tensor product of Hilbert spaces
msc: [46M05, 46C05]
sources:
  - url: https://en.wikipedia.org/wiki/Tensor_product_of_Hilbert_spaces
same_as:
  wikidata: Q1404488
  wikipedia: Tensor product of Hilbert spaces
---

## Summary

The tensor product of two Hilbert spaces is the completion of their algebraic tensor product.
Its inner product multiplies the inner products of the factors.
It is again a Hilbert space, and products of orthonormal basis vectors form an orthonormal basis of it.
In quantum mechanics it describes a system made of two subsystems.
In analysis, $L^2(X) \mathbin{\widehat{\otimes}} L^2(Y)$ is $L^2(X \times Y)$.

## Explanations

### L1 human

If one Hilbert space describes functions of $x$ and another describes functions of $y$, the tensor product describes functions of the pair $(x, y)$.
The simplest elements are products $f(x) g(y)$, and general elements are limits of sums of such products.
For example, $\C^2 \otimes \C^3$ is $\C^6$, with the six products $e_i \otimes f_j$ of basis vectors as an orthonormal basis.