Inductive tensor product

The strongest 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 ι-topology. When X ⊗ Y is endowed with this topology then it is denoted by ${\displaystyle X\otimes _{\iota }Y}$ and called the inductive tensor product of X and Y.[1]

Preliminaries

Throughout let X,Y, and Z be 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 L, has the subspace topology induced by Y.
  • If S is a subspace of 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 ${\displaystyle L_{0}\left(x+\operatorname {ker} L\right):=L(x)}$ 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 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* or ${\displaystyle X^{\prime }}$ and the algebraic dual space (which is the vector space of all linear functionals on 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 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 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 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 U of the origin in X such that ${\displaystyle \Lambda (U)}$ is precompact in 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 H (i = 1, 2, ${\displaystyle \ldots }$) 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 L.[4]

Notation for topologies

• σ(X, X′) denotes the coarsest topology on X making every map in X′ continuous and ${\displaystyle X_{\sigma \left(X,X^{\prime }\right)}}$ or ${\displaystyle X_{\sigma }}$ denotes X endowed with this topology.
• σ(X′, X) denotes weak-* topology on X* and ${\displaystyle X_{\sigma \left(X^{\prime },X\right)}}$ or ${\displaystyle X_{\sigma }^{\prime }}$ denotes X′ endowed with this topology.
  • Note that every ${\displaystyle x_{0}\in X}$ induces a map ${\displaystyle X^{\prime }\to \mathbb {R} }$ defined by ${\displaystyle \lambda \mapsto \lambda \left(x_{0}\right)}$. σ(X′, X) is the coarsest topology on X′ making all such maps continuous.
• b(X, X′) denotes the topology of bounded convergence on X and ${\displaystyle X_{b\left(X,X^{\prime }\right)}}$ or ${\displaystyle X_{b}}$ denotes X endowed with this topology.
• b(X′, X) denotes the topology of bounded convergence on X′ or the strong dual topology on X′ and ${\displaystyle X_{b\left(X^{\prime },X\right)}}$ or ${\displaystyle X_{b}^{\prime }}$ denotes X′ endowed with this topology.
  • As usual, if X* 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′, 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 ${\displaystyle X\times Y\to Z}$, going into the space of all linear mappings of ${\displaystyle X\otimes Y\to Z}$.[1] Then when the domain of I is restricted to ${\displaystyle {\mathcal {B}}\left(X,Y;Z\right)}$ (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 𝜏 is a locally convex TVS topology on X ⊗ Y (X ⊗ Y with this topology will be denoted by ${\displaystyle X\otimes _{\tau }Y}$), then 𝜏 is equal to the inductive tensor product topology if and only if it has the following property:[5]

For every locally convex TVS Z, if 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 I is restricted to ${\displaystyle {\mathcal {B}}\left(X,Y;Z\right)}$ (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 ${\displaystyle X\otimes _{\tau }Y\to Z}$.

See also

• Auxiliary normed spaces
• Initial topology
• Injective tensor product
• Nuclear operators
• Nuclear spaces
• Projective tensor product
• Tensor product of Hilbert spaces
• Topological tensor product

References

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

Bibliography

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• 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. 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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External links

• Nuclear space at ncatlab