Revision as of 09:28, 25 May 2010

The tensor product of Hilbert spaces</math>

Recall that $(e_k)_{k\in\mathbb{N}}$ is the canonical basis of $H=\ell^2(\mathbb{N})$ . The space $H\tens H$ is the collection of sequences $(x_{np})_{n,p\in\mathbb{N}}$ of complex numbers such that:

∑	\| x_np \| ²
n,p

converges. The scalar product is defined just as before:

$\langle (x_{np}), (y_{np})\rangle = \sum_{n,p} x_{np}\bar y_{np}$ .

The canonical basis of $H\tens H$ is denoted $(e_{ij})_{i,j\in\mathbb{N}}$ where $e i j$ is the (doubly indexed) sequence $(e_{ijnp})_{n,p\in\mathbb{N}}$ defined by: