Inductive tensor product

The finest locally convex topological vector space (TVS) topology on [math]\displaystyle{ X \otimes Y, }[/math] the tensor product of two locally convex TVSs, making the canonical map [math]\displaystyle{ \cdot \otimes \cdot : X \times Y \to X \otimes Y }[/math] (defined by sending [math]\displaystyle{ (x, y) \in X \times Y }[/math] to [math]\displaystyle{ x \otimes y }[/math]) separately continuous is called the inductive topology or the [math]\displaystyle{ \iota }[/math]-topology. When [math]\displaystyle{ X \otimes Y }[/math] is endowed with this topology then it is denoted by [math]\displaystyle{ X \otimes_{\iota} Y }[/math] and called the inductive tensor product of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y. }[/math]

Preliminaries

Throughout let [math]\displaystyle{ X, Y, }[/math] and [math]\displaystyle{ Z }[/math] be locally convex topological vector spaces and [math]\displaystyle{ L : X \to Y }[/math] be a linear map.

• [math]\displaystyle{ L : X \to Y }[/math] is a topological homomorphism or homomorphism, if it is linear, continuous, and [math]\displaystyle{ L : X \to \operatorname{Im} L }[/math] is an open map, where [math]\displaystyle{ \operatorname{Im} L, }[/math] the image of [math]\displaystyle{ L, }[/math] has the subspace topology induced by [math]\displaystyle{ Y. }[/math]
  • If [math]\displaystyle{ S \subseteq X }[/math] is a subspace of [math]\displaystyle{ X }[/math] then both the quotient map [math]\displaystyle{ X \to X / S }[/math] and the canonical injection [math]\displaystyle{ S \to X }[/math] are homomorphisms. In particular, any linear map [math]\displaystyle{ L : X \to Y }[/math] can be canonically decomposed as follows: [math]\displaystyle{ X \to X / \operatorname{ker} L \overset{L_0}{\rightarrow} \operatorname{Im} L \to Y }[/math] where [math]\displaystyle{ L_0(x + \ker L) := L(x) }[/math] defines a bijection.
• The set of continuous linear maps [math]\displaystyle{ X \to Z }[/math] (resp. continuous bilinear maps [math]\displaystyle{ X \times Y \to Z }[/math]) will be denoted by [math]\displaystyle{ L(X; Z) }[/math] (resp. [math]\displaystyle{ B(X, Y; Z) }[/math]) where if [math]\displaystyle{ Z }[/math] is the scalar field then we may instead write [math]\displaystyle{ L(X) }[/math] (resp. [math]\displaystyle{ B(X, Y) }[/math]).
• We will denote the continuous dual space of [math]\displaystyle{ X }[/math] by [math]\displaystyle{ X^{\prime} }[/math] and the algebraic dual space (which is the vector space of all linear functionals on [math]\displaystyle{ X, }[/math] whether continuous or not) by [math]\displaystyle{ X^{\#}. }[/math]
  • To increase the clarity of the exposition, we use the common convention of writing elements of [math]\displaystyle{ X^{\prime} }[/math] with a prime following the symbol (e.g. [math]\displaystyle{ x^{\prime} }[/math] denotes an element of [math]\displaystyle{ X^{\prime} }[/math] and not, say, a derivative and the variables [math]\displaystyle{ x }[/math] and [math]\displaystyle{ x^{\prime} }[/math] need not be related in any way).
• A linear map [math]\displaystyle{ L : H \to H }[/math] from a Hilbert space into itself is called positive if [math]\displaystyle{ \langle L(x), X \rangle \geq 0 }[/math] for every [math]\displaystyle{ x \in H. }[/math] In this case, there is a unique positive map [math]\displaystyle{ r : H \to H, }[/math] called the square-root of [math]\displaystyle{ L, }[/math] such that [math]\displaystyle{ L = r \circ r. }[/math]
  • If [math]\displaystyle{ L : H_1 \to H_2 }[/math] is any continuous linear map between Hilbert spaces, then [math]\displaystyle{ L^* \circ L }[/math] is always positive. Now let [math]\displaystyle{ R : H \to H }[/math] denote its positive square-root, which is called the absolute value of [math]\displaystyle{ L. }[/math] Define [math]\displaystyle{ U : H_1 \to H_2 }[/math] first on [math]\displaystyle{ \operatorname{Im} R }[/math] by setting [math]\displaystyle{ U(x) = L(x) }[/math] for [math]\displaystyle{ x = R \left(x_1\right) \in \operatorname{Im} R }[/math] and extending [math]\displaystyle{ U }[/math] continuously to [math]\displaystyle{ \overline{\operatorname{Im} R}, }[/math] and then define [math]\displaystyle{ U }[/math] on [math]\displaystyle{ \operatorname{ker} R }[/math] by setting [math]\displaystyle{ U(x) = 0 }[/math] for [math]\displaystyle{ x \in \operatorname{ker} R }[/math] and extend this map linearly to all of [math]\displaystyle{ H_1. }[/math] The map [math]\displaystyle{ U\big\vert_{\operatorname{Im} R} : \operatorname{Im} R \to \operatorname{Im} L }[/math] is a surjective isometry and [math]\displaystyle{ L = U \circ R. }[/math]
• A linear map [math]\displaystyle{ \Lambda : X \to Y }[/math] is called compact or completely continuous if there is a neighborhood [math]\displaystyle{ U }[/math] of the origin in [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ \Lambda(U) }[/math] is precompact in [math]\displaystyle{ Y. }[/math]
  • In a Hilbert space, positive compact linear operators, say [math]\displaystyle{ L : H \to H }[/math] 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, [math]\displaystyle{ r_1 \gt r_2 \gt \cdots \gt r_k \gt \cdots }[/math] and a sequence of nonzero finite dimensional subspaces [math]\displaystyle{ V_i }[/math] of [math]\displaystyle{ H }[/math] ([math]\displaystyle{ i = 1, 2, \ldots }[/math]) with the following properties: (1) the subspaces [math]\displaystyle{ V_i }[/math] are pairwise orthogonal; (2) for every [math]\displaystyle{ i }[/math] and every [math]\displaystyle{ x \in V_i, }[/math] [math]\displaystyle{ L(x) = r_i x }[/math]; and (3) the orthogonal of the subspace spanned by [math]\displaystyle{ \cup_i V_i }[/math] is equal to the kernel of [math]\displaystyle{ L. }[/math]

Notation for topologies

• [math]\displaystyle{ \sigma\left(X, X^{\prime}\right) }[/math] denotes the coarsest topology on [math]\displaystyle{ X }[/math] making every map in [math]\displaystyle{ X^{\prime} }[/math] continuous and [math]\displaystyle{ X_{\sigma\left(X, X^{\prime}\right)} }[/math] or [math]\displaystyle{ X_{\sigma} }[/math] denotes [math]\displaystyle{ X }[/math] endowed with this topology.
• [math]\displaystyle{ \sigma\left(X^{\prime}, X\right) }[/math] denotes weak-* topology on [math]\displaystyle{ X^{\prime} }[/math] and [math]\displaystyle{ X_{\sigma\left(X^{\prime}, X\right)} }[/math] or [math]\displaystyle{ X^{\prime}_{\sigma} }[/math] denotes [math]\displaystyle{ X^{\prime} }[/math] endowed with this topology.
  • Every [math]\displaystyle{ x_0 \in X }[/math] induces a map [math]\displaystyle{ X^{\prime} \to \R }[/math] defined by [math]\displaystyle{ \lambda \mapsto \lambda \left(x_0\right). }[/math] [math]\displaystyle{ \sigma\left(X^{\prime}, X\right) }[/math] is the coarsest topology on [math]\displaystyle{ X^{\prime} }[/math] making all such maps continuous.
• [math]\displaystyle{ b\left(X, X^{\prime}\right) }[/math] denotes the topology of bounded convergence on [math]\displaystyle{ X }[/math] and [math]\displaystyle{ X_{b\left(X, X^{\prime}\right)} }[/math] or [math]\displaystyle{ X_b }[/math] denotes [math]\displaystyle{ X }[/math] endowed with this topology.
• [math]\displaystyle{ b\left(X^{\prime}, X\right) }[/math] denotes the topology of bounded convergence on [math]\displaystyle{ X^{\prime} }[/math] or the strong dual topology on [math]\displaystyle{ X^{\prime} }[/math] and [math]\displaystyle{ X_{b\left(X^{\prime}, X\right)} }[/math] or [math]\displaystyle{ X^{\prime}_b }[/math] denotes [math]\displaystyle{ X^{\prime} }[/math] endowed with this topology.
  • As usual, if [math]\displaystyle{ X^{\prime} }[/math] 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 [math]\displaystyle{ b\left(X^{\prime}, X\right). }[/math]

Universal property

Suppose that [math]\displaystyle{ Z }[/math] is a locally convex space and that [math]\displaystyle{ I }[/math] is the canonical map from the space of all bilinear mappings of the form [math]\displaystyle{ X \times Y \to Z, }[/math] going into the space of all linear mappings of [math]\displaystyle{ X \otimes Y \to Z. }[/math] Then when the domain of [math]\displaystyle{ I }[/math] is restricted to [math]\displaystyle{ \mathcal{B}(X, Y; Z) }[/math] (the space of separately continuous bilinear maps) then the range of this restriction is the space [math]\displaystyle{ L\left(X \otimes_{\iota} Y; Z\right) }[/math] of continuous linear operators [math]\displaystyle{ X \otimes_{\iota} Y \to Z. }[/math] In particular, the continuous dual space of [math]\displaystyle{ X \otimes_{\iota} Y }[/math] is canonically isomorphic to the space [math]\displaystyle{ \mathcal{B}(X, Y), }[/math] the space of separately continuous bilinear forms on [math]\displaystyle{ X \times Y. }[/math]

If [math]\displaystyle{ \tau }[/math] is a locally convex TVS topology on [math]\displaystyle{ X \otimes Y }[/math] ([math]\displaystyle{ X \otimes Y }[/math] with this topology will be denoted by [math]\displaystyle{ X \otimes_{\tau} Y }[/math]), then [math]\displaystyle{ \tau }[/math] is equal to the inductive tensor product topology if and only if it has the following property:

For every locally convex TVS [math]\displaystyle{ Z, }[/math] if [math]\displaystyle{ I }[/math] is the canonical map from the space of all bilinear mappings of the form [math]\displaystyle{ X \times Y \to Z, }[/math] going into the space of all linear mappings of [math]\displaystyle{ X \otimes Y \to Z, }[/math] then when the domain of [math]\displaystyle{ I }[/math] is restricted to [math]\displaystyle{ \mathcal{B}(X, Y; Z) }[/math] (space of separately continuous bilinear maps) then the range of this restriction is the space [math]\displaystyle{ L\left(X \otimes_{\tau} Y; Z\right) }[/math] of continuous linear operators [math]\displaystyle{ X \otimes_{\tau} Y \to Z. }[/math]

See also

• Initial topology – Coarsest topology making certain functions continuous
• Injective tensor product
• Nuclear operator
• Nuclear space – A generalization of finite dimensional Euclidean spaces different from Hilbert spaces
• Projective tensor product
• Tensor product of Hilbert spaces – Tensor product space endowed with a special inner product
• Topological tensor product – Tensor product constructions for topological vector spaces

Bibliography

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External links

• Nuclear space at ncatlab