The finest locally convex topological vector space (TVS) topology on the tensor product of two locally convex TVSs, making the canonical map (defined by sending to ) separately continuous is called the inductive topology or the -topology. When is endowed with this topology then it is denoted by and called the inductive tensor product of and [1]
Preliminaries
Throughout let and
be locally convex topological vector spaces and
be a linear map.
is a topological homomorphism or homomorphism, if it is linear, continuous, and
is an open map, where
the image of
has the subspace topology induced by
- If
is a subspace of
then both the quotient map
and the canonical injection
are homomorphisms. In particular, any linear map
can be canonically decomposed as follows:
where
defines a bijection.
- If
- The set of continuous linear maps
(resp. continuous bilinear maps
) will be denoted by
(resp.
) where if
is the scalar field then we may instead write
(resp.
).
- We will denote the continuous dual space of
by
and the algebraic dual space (which is the vector space of all linear functionals on
whether continuous or not) by
- To increase the clarity of the exposition, we use the common convention of writing elements of
with a prime following the symbol (e.g.
denotes an element of
and not, say, a derivative and the variables
and
need not be related in any way).
- To increase the clarity of the exposition, we use the common convention of writing elements of
- A linear map
from a Hilbert space into itself is called positive if
for every
In this case, there is a unique positive map
called the square-root of
such that
[2]
- If
is any continuous linear map between Hilbert spaces, then
is always positive. Now let
denote its positive square-root, which is called the absolute value of
Define
first on
by setting
for
and extending
continuously to
and then define
on
by setting
for
and extend this map linearly to all of
The map
is a surjective isometry and
- If
- A linear map
is called compact or completely continuous if there is a neighborhood
of the origin in
such that
is precompact in
[3]
- In a Hilbert space, positive compact linear operators, say
have a simple spectral decomposition discovered at the beginning of the 20th century by Fredholm and F. Riesz:[4]
- In a Hilbert space, positive compact linear operators, say
- There is a sequence of positive numbers, decreasing and either finite or else converging to 0,
and a sequence of nonzero finite dimensional subspaces
of
(
) with the following properties: (1) the subspaces
are pairwise orthogonal; (2) for every
and every
; and (3) the orthogonal of the subspace spanned by
is equal to the kernel of
[4]
- There is a sequence of positive numbers, decreasing and either finite or else converging to 0,
Notation for topologies
denotes the coarsest topology on
making every map in
continuous and
or
denotes
endowed with this topology.
denotes weak-* topology on
and
or
denotes
endowed with this topology.
- Every
induces a map
defined by
is the coarsest topology on
making all such maps continuous.
- Every
denotes the topology of bounded convergence on
and
or
denotes
endowed with this topology.
denotes the topology of bounded convergence on
or the strong dual topology on
and
or
denotes
endowed with this topology.
- As usual, if
is considered as a topological vector space but it has not been made clear what topology it is endowed with, then the topology will be assumed to be
- As usual, if
Universal property
Suppose that is a locally convex space and that
is the canonical map from the space of all bilinear mappings of the form
going into the space of all linear mappings of
[1] Then when the domain of
is restricted to
(the space of separately continuous bilinear maps) then the range of this restriction is the space
of continuous linear operators
In particular, the continuous dual space of
is canonically isomorphic to the space
the space of separately continuous bilinear forms on
If is a locally convex TVS topology on
(
with this topology will be denoted by
), then
is equal to the inductive tensor product topology if and only if it has the following property:[5]
- For every locally convex TVS
if
is the canonical map from the space of all bilinear mappings of the form
going into the space of all linear mappings of
then when the domain of
is restricted to
(space of separately continuous bilinear maps) then the range of this restriction is the space
of continuous linear operators
See also
- Auxiliary normed spaces
- Initial topology – Coarsest topology making certain functions continuous
- Injective tensor product
- Nuclear operator – Linear operator related to topological vector spaces
- Nuclear space – A generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
- Projective tensor product – tensor product defined on two topological vector spaces
- Tensor product of Hilbert spaces – Tensor product space endowed with a special inner product
- Topological tensor product – Tensor product constructions for topological vector spaces
References
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