This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (June 2020) (Learn how and when to remove this message)

This article needs attention from an expert in Mathematics. The specific problem is: It's a specialized topic of Functional analysis. WikiProject Mathematics may be able to help recruit an expert. (April 2020)

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 ${\displaystyle X\otimes _{\tau }Y\to Z.}$

• Auxiliary normed spaces
• Initial topology – Coarsest topology making certain functions continuous
• Injective tensor product
• Nuclear operator – Linear operator related to topological vector spaces
• 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

• 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.

• Nuclear space at ncatlab