In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, there are usually many different ways to construct a topological tensor product of two
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s. For
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s or
nuclear space
In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces ...
s there is a simple
well-behaved
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition,
it is sometimes called well-behaved. Th ...
theory of
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
s (see
Tensor product of Hilbert spaces In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly spea ...
), but for general
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s or
locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
s the theory is notoriously subtle.
Motivation
One of the original motivations for topological tensor products
is the fact that tensor products of the spaces of smooth functions on
do not behave as expected. There is an injection
:
but this is not an isomorphism. For example, the function
cannot be expressed as a finite linear combination of smooth functions in
We only get an isomorphism after constructing the topological tensor product; i.e.,
:
This article first details the construction in the Banach space case.
is not a Banach space and further cases are discussed at the end.
Tensor products of Hilbert spaces
The algebraic tensor product of two Hilbert spaces ''A'' and ''B'' has a natural positive definite
sesquilinear form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...
(scalar product) induced by the sesquilinear forms of ''A'' and ''B''. So in particular it has a natural
positive definite quadratic form
Positive is a property of positivity and may refer to:
Mathematics and science
* Positive formula, a logical formula not containing negation
* Positive number, a number that is greater than 0
* Plus sign, the sign "+" used to indicate a pos ...
, and the corresponding completion is a Hilbert space ''A'' ⊗ ''B'', called the (Hilbert space) tensor product of ''A'' and ''B''.
If the vectors ''a
i'' and ''b
j'' run through
orthonormal bases
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
of ''A'' and ''B'', then the vectors ''a
i''⊗''b
j'' form an orthonormal basis of ''A'' ⊗ ''B''.
Cross norms and tensor products of Banach spaces
We shall use the notation from in this section. The obvious way to define the tensor product of two Banach spaces
and
is to copy the method for Hilbert spaces: define a norm on the algebraic tensor product, then take the completion in this norm. The problem is that there is more than one natural way to define a norm on the tensor product.
If
and
are Banach spaces the algebraic tensor product of
and
means the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
of
and
as vector spaces and is denoted by
The algebraic tensor product
consists of all finite sums
where
is a natural number depending on
and
and
for
When
and
are Banach spaces, a (or )
on the algebraic tensor product
is a norm satisfying the conditions
Here
and
are elements of the
topological dual spaces of
and
respectively, and
is the
dual norm
In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space.
Definition
Let X be a normed vector space with norm \, \cdot\, and let X^* denote its continuous dual space. The dual ...
of
The term is also used for the definition above.
There is a cross norm
called the projective cross norm, given by
where
It turns out that the projective cross norm agrees with the largest cross norm (, proposition 2.1).
There is a cross norm
called the injective cross norm, given by
where
Here
and
denote the topological duals of
and
respectively.
Note hereby that the injective cross norm is only in some reasonable sense the "smallest".
The completions of the algebraic tensor product in these two norms are called the projective and injective tensor products, and are denoted by
and
When
and
are Hilbert spaces, the norm used for their Hilbert space tensor product is not equal to either of these norms in general. Some authors denote it by
so the Hilbert space tensor product in the section above would be
A
is an assignment to each pair
of Banach spaces of a reasonable crossnorm on
so that if
are arbitrary Banach spaces then for all (continuous linear) operators
and
the operator
is continuous and
If
and
are two Banach spaces and
is a uniform cross norm then
defines a reasonable cross norm on the algebraic tensor product
The normed linear space obtained by equipping
with that norm is denoted by
The completion of
which is a Banach space, is denoted by
The value of the norm given by
on
and on the completed tensor product
for an element
in
(or
) is denoted by
A uniform crossnorm
is said to be if, for every pair
of Banach spaces and every
A uniform crossnorm
is if, for every pair
of Banach spaces and every
A is defined to be a finitely generated uniform crossnorm. The projective cross norm
and the injective cross norm
defined above are tensor norms and they are called the projective tensor norm and the injective tensor norm, respectively.
If
and
are arbitrary Banach spaces and
is an arbitrary uniform cross norm then
Tensor products of locally convex topological vector spaces
The topologies of locally convex topological vector spaces
and
are given by families of
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
s. For each choice of seminorm on
and on
we can define the corresponding family of cross norms on the algebraic tensor product
and by choosing one cross norm from each family we get some cross norms on
defining a topology. There are in general an enormous number of ways to do this. The two most important ways are to take all the projective cross norms, or all the injective cross norms. The completions of the resulting topologies on
are called the projective and injective tensor products, and denoted by
and
There is a natural map from
to
If
or
is a
nuclear space
In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces ...
then the natural map from
to
is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. Roughly speaking, this means that if
or
is nuclear, then there is only one sensible tensor product of
and
.
This property characterizes nuclear spaces.
See also
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References
*.
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{{TopologicalTensorProductsAndNuclearSpaces
Operator theory
Topological vector spaces
Hilbert space