The strongestlocally convex topological vector space topology on, the tensor product of two locally convex TVSs, making the canonical map separately continuous is called the inductive topology or the π-topology. When X ⊗ Y is endowed with this topology then it is denoted by and called the inductive tensor product of X and Y.
is a topological homomorphism or homomorphism, if it is linear, continuous, and is an open map, where, the image ofL, has the subspace topology induced by Y.
* If S is a subspace of X 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 L where if Z is the scalar field then we may instead write L.
* 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 L. Define first on by setting for and extending continuously to, and then define U 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 U of the origin in X such that is precompact in Y.
* Note that every induces a map defined by. σ is the coarsest topology on X′ making all such maps continuous.
b denotes the topology of bounded convergence on X and or denotes X endowed with this topology.
b denotes the topology of bounded convergence on X′ or the strong dual topology on X′ and or denotes X′ endowed with this topology.
* As usual, if X* 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 b.
Universal property
Suppose that Z is a locally convex space and that I 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 I 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 X ⊗ Y, then ? is equal to the inductive tensor product topology if and only if it has the following property:
