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.

$L:X\to Y$ $L:X\to Y$ is a topological homomorphism or homomorphism, if it is linear, continuous, and $L:X\to \operatorname {Im} L$ $L:X\to \operatorname {Im} L$ is an open map, where $\operatorname {Im} L,$ $\operatorname {Im} L,$ the image of $L,$ $L,$ has the subspace topology induced by $Y.$ $Y.$
- 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 $X$ $X$ by $X^{\prime }$ $X^{\prime }$ and the algebraic dual space (which is the vector space of all linear functionals on $X,$ $X,$ whether continuous or not) by $X^{\#}.$ $X^{\#}.$
- 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 $L:H\to H$ $L:H\to H$ from a Hilbert space into itself is called positive if $\langle L(x),X\rangle \geq 0$ $\langle L(x),X\rangle \geq 0$ for every $x\in H.$ $x\in H.$ In this case, there is a unique positive map $r:H\to H,$ $r:H\to H,$ called the square-root of $L,$ $L,$ such that $L=r\circ r.$ $L=r\circ r.$
- 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 $\Lambda :X\to Y$ $\Lambda :X\to Y$ is called compact or completely continuous if there is a neighborhood $U$ $U$ of the origin in $X$ $X$ such that $\Lambda (U)$ $\Lambda (U)$ is precompact in $Y.$ $Y.$
- 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 $X^{\prime }$ $X^{\prime }$ and $X_{\sigma \left(X^{\prime },X\right)}$ $X_{\sigma \left(X^{\prime },X\right)}$ or $X_{\sigma }^{\prime }$ $X_{\sigma }^{\prime }$ 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 $X_{b\left(X^{\prime },X\right)}$ $X_{b\left(X^{\prime },X\right)}$ or $X_{b}^{\prime }$ $X_{b}^{\prime }$ 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.

References

1 2 Schaefer & Wolff 1999, p. 96.
↑ Trèves 2006, p. 488.
↑ Trèves 2006, p. 483.
1 2 Trèves 2006, p. 490.
↑ Grothendieck 1966, p. 73.

Bibliography

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External links

Nuclear space at ncatlab

[FOOTNOTESchaeferWolff199996-1] 1 2 Schaefer & Wolff 1999, p. 96.

[FOOTNOTETrèves2006488-2] Trèves 2006, p. 488.

[FOOTNOTETrèves2006483-3] Trèves 2006, p. 483.

[FOOTNOTETrèves2006490-4] 1 2 Trèves 2006, p. 490.

[FOOTNOTEGrothendieck196673-5] Grothendieck 1966, p. 73.

v t e Topological tensor products and nuclear spaces
Basic concepts	Auxiliary normed spaces Nuclear space Tensor product Topological tensor product of Hilbert spaces
Topologies	Inductive tensor product Injective tensor product Projective tensor product
Operators/Maps	Fredholm determinant Fredholm kernel Hilbert–Schmidt operator Hypocontinuity Integral Nuclear between Banach spaces Trace class
Theorems	Grothendieck trace theorem Schwartz kernel theorem