iven two Hilbert spaces
one can form pairs
,
,
and finite linear combinations of such
pairs
for any choice of
,
,
,
.

Definition

(Algebraic tensor
product of Hilbert spaces)
Let
then the quotient
space
is called "(algebraic) tersor product" of spaces
and
.
The equivalence class around
is
denoted

Proposition

(Tensor product properties 1) Let
.

1.
have linear
operations:

2.
,
,
s.t.
.

3. The form
defined
by
has properties of scalar product.

Definition

(Complete tensor product
of Hilbert spaces) Completion of
with respect to
is called "(complete) tensor product" of
and
and denoted
.

Proposition

(Tensor product properties 2)

1.
is a Hilbert space.

2. If
is an (orthonormal) basis in
and
is an (orthonormal) basis in
then
is an (orthonormal) basis in
.

Definition

(Tensor product of bounded
operators) Given two operators
and
we define the tensor product
according to the formula (notation of the proposition
(
Tensor product properties
1
)-2):

Proposition

(Tensor product of function
spaces) If
and
are Hilbert spaces of functions on a domain
then then there is a homeomorphism from
to a linear space of functions on
via the correspondence