Inductive tensor product

Template:Expert needed 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 $x\otimes y$) Template:Em 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 $Y.$Template:Sfn

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 $L:X\to \mathrm{Im}L$ is an open map, where $\mathrm{Im}L,$ 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: $X\to X/\ker L\stackrel{L_0}{\to }\mathrm{Im}L\to Y$ 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 $X^{\mathrm{\#}}.$
  • 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 $⟨L(x),X⟩\ge 0$ 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 $L=r\circ r.$Template:Sfn
  • If $L:H_1\to H_2$ is any continuous linear map between Hilbert spaces, then $L^{\ast }\circ L$ 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 $\mathrm{Im}R$ by setting $U(x)=L(x)$ for $x=R\left(x_1\right)\in \mathrm{Im}R$ and extending $U$ continuously to $\overline{\mathrm{Im}R},$ and then define $U$ on $\ker R$ by setting $U(x)=0$ for $x\in \ker R$ and extend this map linearly to all of $H_1.$ The map $U{|}_{\mathrm{Im}R}:\mathrm{Im}R\to \mathrm{Im}L$ is a surjective isometry and $L=U\circ R.$
• A linear map $\mathrm{\Lambda }:X\to Y$ is called compact or completely continuous if there is a neighborhood $U$ of the origin in $X$ such that $\mathrm{\Lambda }(U)$ is precompact in $Y.$Template:Sfn
  • 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:Template:Sfn

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 $H$ ($i=1,2,\dots$) 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 $L.$Template:Sfn

Notation for topologies

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

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 $X\otimes Y\to Z.$Template:Sfn Then when the domain of $I$ is restricted to $ℬ(X,Y;Z)$ (the space of separately continuous bilinear maps) then the range of this restriction is the space $L(X{\otimes }_{\iota }Y;Z)$ 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 $ℬ(X,Y),$ 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:Template:Sfn

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 $ℬ(X,Y;Z)$ (space of separately continuous bilinear maps) then the range of this restriction is the space $L(X{\otimes }_{\tau }Y;Z)$ of continuous linear operators $X{\otimes }_{\tau }Y\to Z.$

See also

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Bibliography

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• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Narici Beckenstein Topological Vector Spaces
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• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels
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External links

• Nuclear space at ncatlab