Inductive tensor product

The finest locally convex topological vector space (TVS) topology on ${\displaystyle X\otimes Y,}$ the tensor product of two locally convex TVSs, making the canonical map ${\displaystyle \cdot \otimes \cdot :X\times Y\to X\otimes Y}$ (defined by sending ${\displaystyle (x,y)\in X\times Y}$ to ${\displaystyle x\otimes y}$) separately continuous is called the inductive topology or the ${\displaystyle \iota }$-topology. When ${\displaystyle X\otimes Y}$ is endowed with this topology then it is denoted by ${\displaystyle X\otimes _{\iota }Y}$ and called the inductive tensor product of ${\displaystyle X}$ and ${\displaystyle Y.}$¹

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

• ${\displaystyle L:X\to Y}$ is a topological homomorphism or homomorphism, if it is linear, continuous, and ${\displaystyle L:X\to \operatorname {Im} L}$ is an open map, where ${\displaystyle \operatorname {Im} L,}$ the image of ${\displaystyle L,}$ has the subspace topology induced by ${\displaystyle Y.}$
  • If ${\displaystyle S\subseteq X}$ is a subspace of ${\displaystyle X}$ then both the quotient map ${\displaystyle X\to X/S}$ and the canonical injection ${\displaystyle S\to X}$ are homomorphisms. In particular, any linear map ${\displaystyle L:X\to Y}$ can be canonically decomposed as follows: ${\displaystyle X\to X/\operatorname {ker} L{\overset {L_{0}}{\rightarrow }}\operatorname {Im} L\to Y}$ where ${\displaystyle L_{0}(x+\ker L):=L(x)}$ defines a bijection.
• The set of continuous linear maps ${\displaystyle X\to Z}$ (resp. continuous bilinear maps ${\displaystyle X\times Y\to Z}$) will be denoted by ${\displaystyle L(X;Z)}$ (resp. ${\displaystyle B(X,Y;Z)}$) where if ${\displaystyle Z}$ is the scalar field then we may instead write ${\displaystyle L(X)}$ (resp. ${\displaystyle B(X,Y)}$).
• We will denote the continuous dual space of ${\displaystyle X}$ by ${\displaystyle X^{\prime }}$ and the algebraic dual space (which is the vector space of all linear functionals on ${\displaystyle X,}$ whether continuous or not) by ${\displaystyle X^{\#}.}$
  • To increase the clarity of the exposition, we use the common convention of writing elements of ${\displaystyle X^{\prime }}$ with a prime following the symbol (e.g. ${\displaystyle x^{\prime }}$ denotes an element of ${\displaystyle X^{\prime }}$ and not, say, a derivative and the variables ${\displaystyle x}$ and ${\displaystyle x^{\prime }}$ need not be related in any way).
• A linear map ${\displaystyle L:H\to H}$ from a Hilbert space into itself is called positive if ${\displaystyle \langle L(x),X\rangle \geq 0}$ for every ${\displaystyle x\in H.}$ In this case, there is a unique positive map ${\displaystyle r:H\to H,}$ called the square-root of ${\displaystyle L,}$ such that ${\displaystyle L=r\circ r.}$²
  • If ${\displaystyle L:H_{1}\to H_{2}}$ is any continuous linear map between Hilbert spaces, then ${\displaystyle L^{*}\circ L}$ is always positive. Now let ${\displaystyle R:H\to H}$ denote its positive square-root, which is called the absolute value of ${\displaystyle L.}$ Define ${\displaystyle U:H_{1}\to H_{2}}$ first on ${\displaystyle \operatorname {Im} R}$ by setting ${\displaystyle U(x)=L(x)}$ for ${\displaystyle x=R\left(x_{1}\right)\in \operatorname {Im} R}$ and extending ${\displaystyle U}$ continuously to ${\displaystyle {\overline {\operatorname {Im} R}},}$ and then define ${\displaystyle U}$ on ${\displaystyle \operatorname {ker} R}$ by setting ${\displaystyle U(x)=0}$ for ${\displaystyle x\in \operatorname {ker} R}$ and extend this map linearly to all of ${\displaystyle H_{1}.}$ The map ${\displaystyle U{\big \vert }_{\operatorname {Im} R}:\operatorname {Im} R\to \operatorname {Im} L}$ is a surjective isometry and ${\displaystyle L=U\circ R.}$
• A linear map ${\displaystyle \Lambda :X\to Y}$ is called compact or completely continuous if there is a neighborhood ${\displaystyle U}$ of the origin in ${\displaystyle X}$ such that ${\displaystyle \Lambda (U)}$ is precompact in ${\displaystyle Y.}$³
  • In a Hilbert space, positive compact linear operators, say ${\displaystyle L:H\to H}$ have a simple spectral decomposition discovered at the beginning of the 20th century by Fredholm and F. Riesz:⁴

There is a sequence of positive numbers, decreasing and either finite or else converging to 0, ${\displaystyle r_{1}>r_{2}>\cdots >r_{k}>\cdots }$ and a sequence of nonzero finite dimensional subspaces ${\displaystyle V_{i}}$ of ${\displaystyle H}$ (${\displaystyle i=1,2,\ldots }$) with the following properties: (1) the subspaces ${\displaystyle V_{i}}$ are pairwise orthogonal; (2) for every ${\displaystyle i}$ and every ${\displaystyle x\in V_{i},}$ ${\displaystyle L(x)=r_{i}x}$; and (3) the orthogonal of the subspace spanned by ${\displaystyle \cup _{i}V_{i}}$ is equal to the kernel of ${\displaystyle L.}$⁴

• ${\displaystyle \sigma \left(X,X^{\prime }\right)}$ denotes the coarsest topology on ${\displaystyle X}$ making every map in ${\displaystyle X^{\prime }}$ continuous and ${\displaystyle X_{\sigma \left(X,X^{\prime }\right)}}$ or ${\displaystyle X_{\sigma }}$ denotes ${\displaystyle X}$ endowed with this topology.
• ${\displaystyle \sigma \left(X^{\prime },X\right)}$ denotes weak-* topology on ${\displaystyle X^{\prime }}$ and ${\displaystyle X_{\sigma \left(X^{\prime },X\right)}}$ or ${\displaystyle X_{\sigma }^{\prime }}$ denotes ${\displaystyle X^{\prime }}$ endowed with this topology.
  • Every ${\displaystyle x_{0}\in X}$ induces a map ${\displaystyle X^{\prime }\to \mathbb {R} }$ defined by ${\displaystyle \lambda \mapsto \lambda \left(x_{0}\right).}$ ${\displaystyle \sigma \left(X^{\prime },X\right)}$ is the coarsest topology on ${\displaystyle X^{\prime }}$ making all such maps continuous.
• ${\displaystyle b\left(X,X^{\prime }\right)}$ denotes the topology of bounded convergence on ${\displaystyle X}$ and ${\displaystyle X_{b\left(X,X^{\prime }\right)}}$ or ${\displaystyle X_{b}}$ denotes ${\displaystyle X}$ endowed with this topology.
• ${\displaystyle b\left(X^{\prime },X\right)}$ denotes the topology of bounded convergence on ${\displaystyle X^{\prime }}$ or the strong dual topology on ${\displaystyle X^{\prime }}$ and ${\displaystyle X_{b\left(X^{\prime },X\right)}}$ or ${\displaystyle X_{b}^{\prime }}$ denotes ${\displaystyle X^{\prime }}$ endowed with this topology.
  • As usual, if ${\displaystyle X^{\prime }}$ 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 ${\displaystyle b\left(X^{\prime },X\right).}$

Suppose that ${\displaystyle Z}$ is a locally convex space and that ${\displaystyle I}$ is the canonical map from the space of all bilinear mappings of the form ${\displaystyle X\times Y\to Z,}$ going into the space of all linear mappings of ${\displaystyle X\otimes Y\to Z.}$¹ Then when the domain of ${\displaystyle I}$ is restricted to ${\displaystyle {\mathcal {B}}(X,Y;Z)}$ (the space of separately continuous bilinear maps) then the range of this restriction is the space ${\displaystyle L\left(X\otimes _{\iota }Y;Z\right)}$ of continuous linear operators ${\displaystyle X\otimes _{\iota }Y\to Z.}$ In particular, the continuous dual space of ${\displaystyle X\otimes _{\iota }Y}$ is canonically isomorphic to the space ${\displaystyle {\mathcal {B}}(X,Y),}$ the space of separately continuous bilinear forms on ${\displaystyle X\times Y.}$

If ${\displaystyle \tau }$ is a locally convex TVS topology on ${\displaystyle X\otimes Y}$ (${\displaystyle X\otimes Y}$ with this topology will be denoted by ${\displaystyle X\otimes _{\tau }Y}$), then ${\displaystyle \tau }$ is equal to the inductive tensor product topology if and only if it has the following property:⁵

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