Inductive tensor product

Not to be confused with Injective 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

Main articles: Topology of uniform convergence and Mackey topology

• $\sigma \left(X,X^{\prime }\right)$ 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 $X\otimes _{\tau }Y\to Z.$

See also

• Auxiliary normed spaces
• Initial topology – Coarsest topology making certain functions continuous
• Injective tensor product
• Nuclear operator
• Nuclear space – A generalization of finite dimensional Euclidean spaces different from Hilbert spaces
• Projective tensor product – tensor product defined on two topological vector spacesPages displaying wikidata descriptions as a fallback
• Tensor product of Hilbert spaces – Tensor product space endowed with a special inner product
• Topological tensor product – Tensor product constructions for topological vector spaces

Bibliography

• Diestel, Joe (2008). The metric theory of tensor products : Grothendieck's résumé revisited. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-4440-3. OCLC 185095773.
• Dubinsky, Ed (1979). The structure of nuclear Fréchet spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09504-7. OCLC 5126156.
• Grothendieck, Alexander (1966). Produits tensoriels topologiques et espaces nucléaires (in French). Providence: American Mathematical Society. ISBN 0-8218-1216-5. OCLC 1315788.
• Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Nlend, H (1977). Bornologies and functional analysis : introductory course on the theory of duality topology-bornology and its use in functional analysis. Amsterdam New York New York: North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier-North Holland. ISBN 0-7204-0712-5. OCLC 2798822.
• Nlend, H (1981). Nuclear and conuclear spaces : introductory courses on nuclear and conuclear spaces in the light of the duality. Amsterdam New York New York, N.Y: North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier North-Holland. ISBN 0-444-86207-2. OCLC 7553061.
• Pietsch, Albrecht (1972). Nuclear locally convex spaces. Berlin, New York: Springer-Verlag. ISBN 0-387-05644-0. OCLC 539541.
• Robertson, A. P. (1973). Topological vector spaces. Cambridge England: University Press. ISBN 0-521-29882-2. OCLC 589250.
• Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.

External links

• Nuclear space at ncatlab