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 $$x \otimes y$$) 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

 * $\sigma\left(X, X^{\prime}\right)$ 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 $$X^{\prime}_{\sigma}$$ denotes $X^{\prime}$ endowed with this topology.
 * Every $$x_0 \in X$$ induces a map $$X^{\prime} \to \R$$ defined by $$\lambda \mapsto \lambda \left(x_0\right).$$ $$\sigma\left(X^{\prime}, X\right)$$ 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 $$X^{\prime}_b$$ 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 $$X \otimes_{\tau} Y \to Z.$$