Inductive tensor product explained

The finest locally convex topological vector space (TVS) topology on

X ⊗ Y, the tensor product of two locally convex TVSs, making the canonical map ⋅ ⊗ ⋅ : X x Y \to X ⊗ Y (defined by sending (x, y) \in X x Y to x ⊗ y ) continuous is called the inductive topology or the \iota -topology. When X ⊗ Y is endowed with this topology then it is denoted by X ⊗ _{\iota} Y and called the inductive tensor product of X and Y.

Preliminaries

Throughout let

X, Y, and Z be locally convex topological vector spaces and L : X \to Y be a linear map.L : X \to Y is a topological homomorphism or homomorphism, if it is linear, continuous, and L : X \to \operatorname{Im} L is an open map, where \operatorname{Im} L, the image of L, has the subspace topology induced by Y.

• • If

S \subseteq X is a subspace of X then both the quotient map X \to X / S and the canonical injection S \to X are homomorphisms. In particular, any linear map L : X \to Y can be canonically decomposed as follows: X \to X / \operatorname{ker} L \overset{L _{0}{ → }} \operatorname{Im} L \to Y where L _{0(x} + \ker L) : = L(x) defines a bijection.

• The set of continuous linear maps

X \to Z (resp. continuous bilinear maps X x Y \to Z ) will be denoted by L(X; Z) (resp. B(X, Y; Z) ) where if Z is the scalar field then we may instead write L(X) (resp. B(X, Y) ).

• We will denote the continuous dual space of

X by X ^{\prime} and the algebraic dual space (which is the vector space of all linear functionals on X, whether continuous or not) by X ^{\#} .

• • To increase the clarity of the exposition, we use the common convention of writing elements of

X ^{\prime} with a prime following the symbol (e.g. x ^{\prime} denotes an element of X ^{\prime} and not, say, a derivative and the variables x and x ^{\prime} need not be related in any way).

• A linear map

L : H \to H from a Hilbert space into itself is called positive if \langle L(x), X \rangle \geq 0 for every x \in H. In this case, there is a unique positive map r : H \to H, called the square-root of L, such that L = r \circ r.

• • If

L : H _{1} \to H _{2} is any continuous linear map between Hilbert spaces, then L ^{*} \circ L is always positive. Now let R : H \to H denote its positive square-root, which is called the absolute value of L. Define U : H _{1} \to H _{2} first on \operatorname{Im} R by setting U(x) = L(x) for x = R \left(x _{1\right)} \in \operatorname{Im} R and extending U continuously to \overline{\operatorname{Im} R}, and then define U on \operatorname{ker} R by setting U(x) = 0 for x \in \operatorname{ker} R and extend this map linearly to all of H _{1.} The map U \ vert _{\operatorname{Im} R} : \operatorname{Im} R \to \operatorname{Im} L is a surjective isometry and L = U \circ R.

• A linear map

Λ : X \to Y is called compact or completely continuous if there is a neighborhood U of the origin in X such that Λ (U) is precompact in Y.

• • In a Hilbert space, positive compact linear operators, say

L : H \to H have a simple spectral decomposition discovered at the beginning of the 20th century by Fredholm and F. Riesz:

There is a sequence of positive numbers, decreasing and either finite or else converging to 0,

r _{1} > r _{2} > … > r _{k} > … and a sequence of nonzero finite dimensional subspaces V _{i} of H (i = 1 , 2 , \ldots ) with the following properties: (1) the subspaces V _{i} are pairwise orthogonal; (2) for every i and every x \in V _{i,} L(x) = r _{i} x ; and (3) the orthogonal of the subspace spanned by \cup _{i} V _{i} is equal to the kernel of L.

Notation for topologies

See main article: Mackey topology.

• \sigma\left(X, X ^{\prime} \right) denotes the coarsest topology on

X making every map in X ^{\prime} continuous and X _{\sigma\left(X, X ^{\prime} \right)} or X _{\sigma} denotes X endowed with this topology.

• \sigma\left(X ^{\prime} , X\right) denotes weak-* topology on

X ^{\prime} and X _{\sigma\left(X ^{\prime} , X\right)} or X ^{\prime} _{\sigma} denotes X ^{\prime} endowed with this topology.

• • Every

x _{0} \in X induces a map X ^{\prime} \to \R defined by λ \mapsto λ \left(x _{0\right).} \sigma\left(X ^{\prime} , X\right) is the coarsest topology on X ^{\prime} making all such maps continuous.

• b\left(X, X ^{\prime} \right) denotes the topology of bounded convergence on

X and X _{b\left(X, X ^{\prime} \right)} or X _{b} denotes X endowed with this topology.

• b\left(X ^{\prime} , X\right) denotes the topology of bounded convergence on

X ^{\prime} or the strong dual topology on X ^{\prime} and X _{b\left(X ^{\prime} , X\right)} or X ^{\prime} _{b} denotes X ^{\prime} endowed with this topology.

• • As usual, if

X ^{\prime} 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\left(X ^{\prime} , X\right).

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 X x Y \to Z, going into the space of all linear mappings of X ⊗ Y \to Z. Then when the domain of I is restricted to l{B}(X, Y; Z) (the space of separately continuous bilinear maps) then the range of this restriction is the space L\left(X ⊗ _{\iota} Y; Z\right) of continuous linear operators X ⊗ _{\iota} Y \to Z. In particular, the continuous dual space of X ⊗ _{\iota} Y is canonically isomorphic to the space l{B}(X, Y), the space of separately continuous bilinear forms on X x Y.

If

\tau is a locally convex TVS topology on X ⊗ Y (X ⊗ Y with this topology will be denoted by X ⊗ _{\tau} Y ), then \tau is equal to the inductive tensor product topology if and only if it has the following property:

For every locally convex TVS

Z, if I is the canonical map from the space of all bilinear mappings of the form X x Y \to Z, going into the space of all linear mappings of X ⊗ Y \to Z, then when the domain of I is restricted to l{B}(X, Y; Z) (space of separately continuous bilinear maps) then the range of this restriction is the space L\left(X ⊗ _{\tau} Y; Z\right) of continuous linear operators X ⊗ _{\tau} Y \to Z.

Bibliography

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External links

• Nuclear space at ncatlab