Inductive tensor product

The finest locally convex topological vector space topology on the tensor product of two locally convex TVSs, making the canonical map 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

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.
The set of continuous linear maps will be denoted by where if is the scalar field then we may instead write .
We will denote the continuous dual space of by and the algebraic dual space by
* To increase the clarity of the exposition, we use the common convention of writing elements of with a prime following the symbol.
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
* 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
A linear map is called compact or completely continuous if there is a neighborhood of the origin in such that is precompact in
* 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:

Notation for topologies

[Topology of uniform convergence#The weak topology σ(X, X')|] denotes the coarsest topology on making every map in continuous and or denotes endowed with this topology.
[Topology of uniform convergence#The weak topology σ(X', X) or the weak* 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.
[Topology of uniform convergence#Bounded convergence b(X, X')|] denotes the topology of bounded convergence on and or denotes endowed with this topology.
[Topology of uniform convergence#Strong dual topology b(X', X)|] 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

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
Then when the domain of is restricted to 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, then is equal to the inductive tensor product topology if and only if it has the following property: