Inductive Tensor Product

The finest locally convex topological vector space (TVS) topology on $X \otimes Y,$ the tensor product of two locally convex TVSs, making the canonical map $\cdot \otimes \cdot : X \times Y \to X \otimes Y$ (defined by sending $(x, y) \in X \times Y$ to $x \otimes y$) continuous is called the inductive topology or the $\iota$-topology. When $X \otimes Y$ is endowed with this topology then it is denoted by $X \otimes_ Y$ 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 $L : X \to \operatorname L$ is an open map, where $\operatorname L,$ 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: $X \to X / \operatorname L \overset \operatorname L \to Y$ where $L_0(x + \ker L) := L(x)$ defines a bijection.
* The set of continuous linear maps $X \to Z$ (resp. continuous bilinear maps $X \times Y \to Z$) 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 $X^$ 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 $X^$ with a prime following the symbol (e.g. $x^$ denotes an element of $X^$ and not, say, a derivative and the variables $x$ and $x^$ 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 $\operatorname R$ by setting $U(x) = L(x)$ for $x = R \left(x_1\right) \in \operatorname R$ and extending $U$ continuously to $\overline,$ and then define $U$ on $\operatorname R$ by setting $U(x) = 0$ for $x \in \operatorname R$ and extend this map linearly to all of $H_1.$ The map $U\big\vert_ : \operatorname R \to \operatorname L$ is a surjective isometry and $L = U \circ R.$
* A linear map $\Lambda : X \to Y$ is called compact or completely continuous if there is a neighborhood $U$ of the origin in $X$ such that $\Lambda(U)$ 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, $r_1 > r_2 > \cdots > r_k > \cdots$ 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

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

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 $X \times Y \to Z,$ going into the space of all linear mappings of $X \otimes Y \to Z.$
Then when the domain of $I$ is restricted to $\mathcal(X, Y; Z)$ (the space of separately continuous bilinear maps) then the range of this restriction is the space $L\left(X \otimes_ Y; Z\right)$ of continuous linear operators $X \otimes_ Y \to Z.$
In particular, the continuous dual space of $X \otimes_ Y$ is canonically isomorphic to the space $\mathcal(X, Y),$ the space of separately continuous bilinear forms on $X \times Y.$

If $\tau$ is a locally convex TVS topology on $X \otimes Y$ ($X \otimes Y$ with this topology will be denoted by $X \otimes_ Y$), 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 $X \times Y \to Z,$ going into the space of all linear mappings of $X \otimes Y \to Z,$ then when the domain of $I$ is restricted to $\mathcal(X, Y; Z)$ (space of separately continuous bilinear maps) then the range of this restriction is the space $L\left(X \otimes_ Y; Z\right)$ of continuous linear operators $X \otimes_ Y \to Z.$

See also

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References

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External links

Nuclear space at ncatlab

{{Functional analysis

Functional analysis
Topological vector spaces
Topology
Topological tensor products