# 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 $x\otimes y$) separately continuous is called the inductive topology or the $\iota$-topology. When $X\otimes Y$ is endowed with this topology then it is denoted by $X\otimes _{\iota }Y$ 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: $X\to X/\operatorname {ker} L{\overset {L_{0}}{\rightarrow }}\operatorname {Im} L\to Y$ where $L_{0}(x+\ker L):=L(x)$ defines a bijection.
* The set of continuous linear maps $X\to Z$ (resp. continuous bilinear maps $X\times Y\to Z$) 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 $X^{\prime }$ 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 $U{\big \vert }_{\operatorname {Im} R}:\operatorname {Im} R\to \operatorname {Im} L$ 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

* $\sigma \left(X,X^{\prime }\right)$ 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 $X_{\sigma }^{\prime }$ denotes $X^{\prime }$ endowed with this topology.
** Every $x_{0}\in X$ induces a map $X^{\prime }\to \mathbb {R}$ defined by $\lambda \mapsto \lambda \left(x_{0}\right).$ $\sigma \left(X^{\prime },X\right)$ 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 $X_{b}^{\prime }$ 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 $X\times Y\to Z,$ going into the space of all linear mappings of $X\otimes Y\to Z.$ Then when the domain of $I$ is restricted to ${\mathcal {B}}(X,Y;Z)$ (the space of separately continuous bilinear maps) then the range of this restriction is the space $L\left(X\otimes _{\iota }Y;Z\right)$ of continuous linear operators $X\otimes _{\iota }Y\to Z.$ In particular, the continuous dual space of $X\otimes _{\iota }Y$ is canonically isomorphic to the space ${\mathcal {B}}(X,Y),$ 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 $X\otimes _{\tau }Y$), 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 $X\times Y\to Z,$ going into the space of all linear mappings of $X\otimes Y\to Z,$ then when the domain of $I$ is restricted to ${\mathcal {B}}(X,Y;Z)$ (space of separately continuous bilinear maps) then the range of this restriction is the space $L\left(X\otimes _{\tau }Y;Z\right)$ of continuous linear operators $X\otimes _{\tau }Y\to Z.$