Английская Википедия:Inductive tensor product

Материал из Онлайн справочника
Перейти к навигацииПерейти к поиску

Шаблон:Distinguish Шаблон:More footnotes needed

Шаблон:Expert needed The finest locally convex topological vector space (TVS) topology on <math>X \otimes Y,</math> the tensor product of two locally convex TVSs, making the canonical map <math>\cdot \otimes \cdot : X \times Y \to X \otimes Y</math> (defined by sending <math>(x, y) \in X \times Y</math> to <math>x \otimes y</math>) Шаблон:Em continuous is called the inductive topology or the <math>\iota</math>-topology. When <math>X \otimes Y</math> is endowed with this topology then it is denoted by <math>X \otimes_{\iota} Y</math> and called the inductive tensor product of <math>X</math> and <math>Y.</math>Шаблон:Sfn

Preliminaries

Throughout let <math>X, Y,</math> and <math>Z</math> be locally convex topological vector spaces and <math>L : X \to Y</math> be a linear map.

  • <math>L : X \to Y</math> is a topological homomorphism or homomorphism, if it is linear, continuous, and <math>L : X \to \operatorname{Im} L</math> is an open map, where <math>\operatorname{Im} L,</math> the image of <math>L,</math> has the subspace topology induced by <math>Y.</math>
    • If <math>S \subseteq X</math> is a subspace of <math>X</math> then both the quotient map <math>X \to X / S</math> and the canonical injection <math>S \to X</math> are homomorphisms. In particular, any linear map <math>L : X \to Y</math> can be canonically decomposed as follows: <math>X \to X / \operatorname{ker} L \overset{L_0}{\rightarrow} \operatorname{Im} L \to Y</math> where <math>L_0(x + \ker L) := L(x)</math> defines a bijection.
  • The set of continuous linear maps <math>X \to Z</math> (resp. continuous bilinear maps <math>X \times Y \to Z</math>) will be denoted by <math>L(X; Z)</math> (resp. <math>B(X, Y; Z)</math>) where if <math>Z</math> is the scalar field then we may instead write <math>L(X)</math> (resp. <math>B(X, Y)</math>).
  • We will denote the continuous dual space of <math>X</math> by <math>X^{\prime}</math> and the algebraic dual space (which is the vector space of all linear functionals on <math>X,</math> whether continuous or not) by <math>X^{\#}.</math>
    • To increase the clarity of the exposition, we use the common convention of writing elements of <math>X^{\prime}</math> with a prime following the symbol (e.g. <math>x^{\prime}</math> denotes an element of <math>X^{\prime}</math> and not, say, a derivative and the variables <math>x</math> and <math>x^{\prime}</math> need not be related in any way).
  • A linear map <math>L : H \to H</math> from a Hilbert space into itself is called positive if <math>\langle L(x), X \rangle \geq 0</math> for every <math>x \in H.</math> In this case, there is a unique positive map <math>r : H \to H,</math> called the square-root of <math>L,</math> such that <math>L = r \circ r.</math>Шаблон:Sfn
    • If <math>L : H_1 \to H_2</math> is any continuous linear map between Hilbert spaces, then <math>L^* \circ L</math> is always positive. Now let <math>R : H \to H</math> denote its positive square-root, which is called the absolute value of <math>L.</math> Define <math>U : H_1 \to H_2</math> first on <math>\operatorname{Im} R</math> by setting <math>U(x) = L(x)</math> for <math>x = R \left(x_1\right) \in \operatorname{Im} R</math> and extending <math>U</math> continuously to <math>\overline{\operatorname{Im} R},</math> and then define <math>U</math> on <math>\operatorname{ker} R</math> by setting <math>U(x) = 0</math> for <math>x \in \operatorname{ker} R</math> and extend this map linearly to all of <math>H_1.</math> The map <math>U\big\vert_{\operatorname{Im} R} : \operatorname{Im} R \to \operatorname{Im} L</math> is a surjective isometry and <math>L = U \circ R.</math>
  • A linear map <math>\Lambda : X \to Y</math> is called compact or completely continuous if there is a neighborhood <math>U</math> of the origin in <math>X</math> such that <math>\Lambda(U)</math> is precompact in <math>Y.</math>Шаблон:Sfn
    • In a Hilbert space, positive compact linear operators, say <math>L : H \to H</math> have a simple spectral decomposition discovered at the beginning of the 20th century by Fredholm and F. Riesz:Шаблон:Sfn
There is a sequence of positive numbers, decreasing and either finite or else converging to 0, <math>r_1 > r_2 > \cdots > r_k > \cdots</math> and a sequence of nonzero finite dimensional subspaces <math>V_i</math> of <math>H</math> (<math>i = 1, 2, \ldots</math>) with the following properties: (1) the subspaces <math>V_i</math> are pairwise orthogonal; (2) for every <math>i</math> and every <math>x \in V_i,</math> <math>L(x) = r_i x</math>; and (3) the orthogonal of the subspace spanned by <math>\cup_i V_i</math> is equal to the kernel of <math>L.</math>Шаблон:Sfn

Notation for topologies

Шаблон:Main

Universal property

Suppose that <math>Z</math> is a locally convex space and that <math>I</math> is the canonical map from the space of all bilinear mappings of the form <math>X \times Y \to Z,</math> going into the space of all linear mappings of <math>X \otimes Y \to Z.</math>Шаблон:Sfn Then when the domain of <math>I</math> is restricted to <math>\mathcal{B}(X, Y; Z)</math> (the space of separately continuous bilinear maps) then the range of this restriction is the space <math>L\left(X \otimes_{\iota} Y; Z\right)</math> of continuous linear operators <math>X \otimes_{\iota} Y \to Z.</math> In particular, the continuous dual space of <math>X \otimes_{\iota} Y</math> is canonically isomorphic to the space <math>\mathcal{B}(X, Y),</math> the space of separately continuous bilinear forms on <math>X \times Y.</math>

If <math>\tau</math> is a locally convex TVS topology on <math>X \otimes Y</math> (<math>X \otimes Y</math> with this topology will be denoted by <math>X \otimes_{\tau} Y</math>), then <math>\tau</math> is equal to the inductive tensor product topology if and only if it has the following property:Шаблон:Sfn

For every locally convex TVS <math>Z,</math> if <math>I</math> is the canonical map from the space of all bilinear mappings of the form <math>X \times Y \to Z,</math> going into the space of all linear mappings of <math>X \otimes Y \to Z,</math> then when the domain of <math>I</math> is restricted to <math>\mathcal{B}(X, Y; Z)</math> (space of separately continuous bilinear maps) then the range of this restriction is the space <math>L\left(X \otimes_{\tau} Y; Z\right)</math> of continuous linear operators <math>X \otimes_{\tau} Y \to Z.</math>

See also

References

Шаблон:Reflist

Bibliography

External links

Шаблон:TopologicalTensorProductsAndNuclearSpaces Шаблон:Functional analysis

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:Inductive_tensor_product&oldid=15407198