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strongest
"Strongest" is a song recorded by Norwegian singer and songwriter Ina Wroldsen. The song was released on 27 October 2017 and has peaked at number 2 in Norway.
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locally convex topological vector space (TVS) topology on $X\; \backslash otimes\; Y,$ the tensor product of two locally convex TVSs, making the canonical map $\backslash cdot\; \backslash otimes\; \backslash cdot\; :\; X\; \backslash times\; Y\; \backslash to\; X\; \backslash otimes\; Y$ (defined by sending $(x,\; y)\; \backslash in\; X\; \backslash times\; Y$ to $x\; \backslash otimes\; y$) continuous is called the projective topology or the π-topology. When $X\; \backslash otimes\; Y$ is endowed with this topology then it is denoted by $X\; \backslash otimes\_\; Y$ and called the projective tensor product of $X$ and $Y.$
Preliminaries

Throughout let $X,\; Y,$ and $Z$ be topological vector spaces and $L\; :\; X\; \backslash to\; Y$ be a linear map. * $L\; :\; X\; \backslash to\; Y$ is a topological homomorphism or homomorphism, if it is linear, continuous, and $L\; :\; X\; \backslash to\; \backslash operatorname\; L$ is an open map, where $\backslash operatorname\; L,$ the image of $L,$ has the subspace topology induced by $Y.$ ** If $S\; \backslash subseteq\; X$ is a subspace of $X$ then both the quotient map $X\; \backslash to\; X\; /\; S$ and the canonical injection $S\; \backslash to\; X$ are homomorphisms. In particular, any linear map $L\; :\; X\; \backslash to\; Y$ can be canonically decomposed as follows: $X\; \backslash to\; X\; /\; \backslash operatorname\; L\; \backslash overset\; \backslash operatorname\; L\; \backslash to\; Y$ where $L\_0(x\; +\; \backslash ker\; L)\; :=\; L(x)$ defines a bijection. * The set of continuous linear maps $X\; \backslash to\; Z$ (resp. continuous bilinear maps $X\; \backslash times\; Y\; \backslash to\; Z$) 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 $X^$ 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 $X^$ with a prime following the symbol (e.g. $x^$ denotes an element of $X^$ and not, say, a derivative and the variables $x$ and $x^$ need not be related in any way). * A linear map $L\; :\; H\; \backslash to\; H$ from a Hilbert space into itself is called positive if $\backslash langle\; L(x),\; X\; \backslash rangle\; \backslash geq\; 0$ for every $x\; \backslash in\; H.$ In this case, there is a unique positive map $r\; :\; H\; \backslash to\; H,$ called the square-root of $L,$ such that $L\; =\; r\; \backslash circ\; r.$ ** If $L\; :\; H\_1\; \backslash to\; H\_2$ is any continuous linear map between Hilbert spaces, then $L^*\; \backslash circ\; L$ is always positive. Now let $R\; :\; H\; \backslash to\; H$ denote its positive square-root, which is called the absolute value of $L.$ Define $U\; :\; H\_1\; \backslash to\; H\_2$ first on $\backslash operatorname\; R$ by setting $U(x)\; =\; L(x)$ for $x\; =\; R\; \backslash left(x\_1\backslash right)\; \backslash in\; \backslash operatorname\; R$ and extending $U$ continuously to $\backslash overline,$ and then define $U$ on $\backslash operatorname\; R$ by setting $U(x)\; =\; 0$ for $x\; \backslash in\; \backslash operatorname\; R$ and extend this map linearly to all of $H\_1.$ The map $U\backslash big\backslash vert\_\; :\; \backslash operatorname\; R\; \backslash to\; \backslash operatorname\; L$ is a surjective isometry and $L\; =\; U\; \backslash circ\; R.$ * A linear map $\backslash Lambda\; :\; X\; \backslash to\; Y$ is called compact or completely continuous if there is a neighborhood $U$ of the origin in $X$ such that $\backslash Lambda(U)$ is Totally bounded space#In topological groups, precompact in $Y.$ ** In a Hilbert space, positive compact linear operators, say $L\; :\; H\; \backslash to\; H$ 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, $r\_1\; >\; r\_2\; >\; \backslash cdots\; >\; r\_k\; >\; \backslash cdots$ and a sequence of nonzero finite dimensional subspaces $V\_i$ of $H$ ($i\; =\; 1,\; 2,\; \backslash ldots$) with the following properties: (1) the subspaces $V\_i$ are pairwise orthogonal; (2) for every $i$ and every $x\; \backslash in\; V\_i,$ $L(x)\; =\; r\_i\; x$; and (3) the orthogonal of the subspace spanned by $\backslash cup\_i\; V\_i$ is equal to the kernel of $L.$Notation for topologies

* Topology of uniform convergence#The weak topology σ(X, X'), $\backslash sigma\backslash left(X,\; X^\backslash right)$ denotes the coarsest topology on $X$ making every map in $X^$ continuous and $X\_$ or $X\_$ denotes Topology of uniform convergence#The weak topology σ(X, X'), $X$ endowed with this topology. * Topology of uniform convergence#The weak topology σ(X', X) or the weak* topology, $\backslash sigma\backslash left(X^,\; X\backslash right)$ denotes weak-* topology on $X^$ and $X\_$ or $X^\_$ denotes Topology of uniform convergence#The weak topology σ(X′, X) or the weak* topology, $X^$ endowed with this topology. ** Every $x\_0\; \backslash in\; X$ induces a map $X^\; \backslash to\; \backslash R$ defined by $\backslash lambda\; \backslash mapsto\; \backslash lambda\; \backslash left(x\_0\backslash right).$ $\backslash sigma\backslash left(X^,\; X\backslash right)$ is the coarsest topology on $X^$ making all such maps continuous. * Topology of uniform convergence#Bounded convergence b(X, X'), $b\backslash left(X,\; X^\backslash right)$ denotes the topology of bounded convergence on $X$ and $X\_$ or $X\_b$ denotes Topology of uniform convergence#Bounded convergence b(X, X'), $X$ endowed with this topology. * Topology of uniform convergence#Strong dual topology b(X', X), $b\backslash left(X^,\; X\backslash right)$ denotes the topology of bounded convergence on $X^$ or the strong dual topology on $X^$ and $X\_$ or $X^\_b$ denotes Topology of uniform convergence#Strong dual topology b(X', X), $X^$ 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 $b\backslash left(X^,\; X\backslash right).$A canonical tensor product as a subspace of the dual of Bi(X, Y)

Let $X$ and $Y$ be vector spaces (no topology is needed yet) and let $\backslash operatorname(X,\; Y)$ be the space of all bilinear maps defined on $X\; \backslash times\; Y$ and going into the underlying scalar field. For every $(x,\; y)\; \backslash in\; X\; \backslash times\; Y$ define a canonical bilinear form by $\backslash chi\_$ with domain $\backslash operatorname(X,\; Y)$ by $\backslash chi\_(u)\; :=\; u(x,\; y)$ for every $u\; \backslash in\; Bi(X,\; Y).$ This induces a canonical map $\backslash chi\; :\; X\; \backslash times\; Y\; \backslash to\; \backslash operatorname(X,\; Y)^$ defined by $\backslash chi(x,\; y)\; =\; \backslash chi\_,$ where $\backslash operatorname(X,\; Y)^$ denotes the algebraic dual of $\backslash operatorname(X,\; Y).$ If we denote the span of the range of $\backslash chi$ by $X\; \backslash otimes\; Y$ then $X\; \backslash otimes\; Y$ together with $\backslash chi$ forms a tensor product of $X$ and $Y$ (where $x\; \backslash otimes\; y\; =\; \backslash chi(x,\; y)$). This gives us a canonical tensor product of $X$ and $Y.$ If $Z$ is any other vector space then the mapping $Li(X\; \backslash otimes\; Y;\; Z)\; \backslash to\; \backslash operatorname(X,\; Y;\; Z)$ given by $u\; \backslash mapsto\; u\; \backslash circ\; \backslash chi$ is an isomorphism of vector spaces. In particular, this allows us to identify the algebraic dual of $X\; \backslash otimes\; Y$ with the space of bilinear forms on $X\; \backslash times\; Y.$ Moreover, if $X$ and $Y$ are locally convex topological vector spaces (TVSs) and if $X\; \backslash otimes\; Y$ is given the $\backslash pi$-topology then for every locally convex TVS $Z,$ this map restricts to a vector space isomorphism $L(X\; \backslash otimes\_\; Y;\; Z)\; \backslash to\; B(X,\; Y;\; Z)$ from the space of ''continuous'' linear mappings onto the space of bilinear mappings. In particular, the continuous dual of $X\; \backslash otimes\; Y$ can be canonically identified with the space $\backslash operatorname(X,\; Y)$ of continuous bilinear forms on $X\; \backslash times\; Y$; furthermore, under this identification the equicontinuous subsets of $\backslash operatorname(X,\; Y)$ are the same as the equicontinuous subsets of ''$(X\; \backslash otimes\_\; Y)^.$The projective tensor product

Tensor product of seminorms

Throughout we will let $X$ and $Y$ be locally convex topological vector spaces (local convexity allows us to define useful topologies). If $p$ is a seminorm on $X$ then $C\_p\; :=\; \backslash $ will be its closed unit ball. If $p$ is a seminorm on $X$ and $q$ is a seminorm on $Y$ then we can define the tensor product of $p$ and $q$ to be the map $p\; \backslash otimes\; q$ defined on $X\; \backslash otimes\; Y$ by $$(p\; \backslash otimes\; q)(b)\; :=\; \backslash inf\_\; r$$ where $W$ is the balanced convex hull of $C\_p\; \backslash otimes\; C\_q\; =\; \backslash left\backslash .$ Given $b$ in $X\; \backslash otimes\; Y,$ this can also be expressed as $$(p\; \backslash otimes\; q)(b)\; :=\; \backslash inf\; \backslash sum\_i\; p(x\_i)\; q(y\_i)$$ where the infimum is taken over all finite sequences $x\_1,\; \backslash ldots,\; x\_n\; \backslash in\; X$ and $y\_1,\; \backslash ldots,\; y\_n\; \backslash in\; Y$ (of the same length) such that $b\; =\; x\_1\; \backslash otimes\; y\_1\; +\; \backslash cdots\; +\; x\_n\; \backslash otimes\; y\_n$ (recall that it may not be possible to express $b$ as a simple tensor). If $b\; =\; x\; \backslash otimes\; y$ then we have $$(p\; \backslash otimes\; q)(x\; \backslash otimes\; y)\; =\; p(x)\; q(y).$$ The seminorm $p\; \backslash otimes\; q$ is a norm if and only if both $p$ and $q$ are norms. If the topology of $X$ (resp. $Y$) is given by the family of seminorms $\backslash left(p\_\backslash right)$ (resp. $\backslash left(q\_\backslash right)$) then $X\; \backslash otimes\_\; Y$ is a locally convex space whose topology is given by the family of all possible tensor products of the two families (i.e. by $\backslash left(p\_\; \backslash otimes\; q\_\backslash right)$). In particular, if $X$ and $Y$ are seminormed spaces with seminorms $p$ and $q,$ respectively, then $X\; \backslash otimes\_\; Y$ is a seminormable space whose topology is defined by the seminorm $p\; \backslash otimes\; q.$ If $(X,\; p)$ and $(Y,\; q)$ are normed spaces then $\backslash left(X\; \backslash otimes\; Y,\; p\; \backslash otimes\; q\backslash right)$ is also a normed space, called the projective tensor product of $(X,\; p)$ and $(Y,\; q),$ where the topology induced by $p\; \backslash otimes\; q$ is the same as the π-topology. If $W$ is a convex subset of $X\; \backslash otimes\; Y$ then $W$ is a neighborhood of 0 in $X\; \backslash otimes\_\; Y$ if and only if the preimage of $W$ under the map $(x,\; y)\; \backslash mapsto\; x\; \backslash otimes\; y$ is a neighborhood of 0; equivalent, if and only if there exist open subsets $U\; \backslash subseteq\; X$ and $V\; \backslash subseteq\; Y$ such that this preimage contains $U\; \backslash otimes\; V\; :=\; \backslash .$ It follows that if $\backslash left(U\_\backslash right)$ and $\backslash left(V\_\backslash right)$ are neighborhood bases of the origin in $X$ and $Y,$ respectively, then the set of convex hulls of all possible set $U\_\; \backslash otimes\; V\_$ form a neighborhood basis of the origin in $X\; \backslash otimes\_\; Y.$Universal property

If $\backslash tau$ is a locally convex TVS topology on $X\; \backslash otimes\; Y$ ($X\; \backslash otimes\; Y$ with this topology will be denoted by $X\; \backslash otimes\_\; Y$), then $\backslash tau$ is equal to the π-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 $X\; \backslash times\; Y\; \backslash to\; Z,$ going into the space of all linear mappings of $X\; \backslash otimes\; Y\; \backslash to\; Z,$ then when the domain of $I$ is restricted to $B\backslash left(X,\; Y;\; Z\backslash right)$ then the range of this restriction is the space $L\backslash left(X\; \backslash otimes\_\; Y;\; Z\backslash right)$ of continuous linear operators $X\; \backslash otimes\_\; Y\; \backslash to\; Z.$ In particular, the continuous dual space of $X\; \backslash otimes\_\; Y$ is canonically isomorphic to the space $B(X,\; Y),$ the space of continuous bilinear forms on $X\; \backslash times\; Y.$The π-topology

Note that the canonical vector space isomorphism $I\; :\; B\backslash left(X\_^,\; Y\_^;\; Z\backslash right)\; \backslash to\; L\backslash left(X\; \backslash otimes\_\; Y;\; Z\backslash right)$ preserves equicontinuous subsets. Since $B(X,\; Y)$ is canonically isomorphic to the continuous dual of $X\; \backslash otimes\_\; Y,$ place on $X\; \backslash otimes\; Y$ the topology of uniform convergence on equicontinuous subsets of $\backslash left(X\; \backslash otimes\_\; Y\backslash right)^$; this topology is identical to the π-topology.Preserved properties

Let $X$ and $Y$ be locally convex TVSs. * If both $X$ and $Y$ are Hausdorff (resp. locally convex, Metrizable topological vector space, metrizable, semi-metrizable, Normable space, normable, semi-normable) then so is $X\; \backslash otimes\_\; Y.$Completion

In general, the space $X\; \backslash otimes\_\; Y$ is not complete, even if both $X$ and $Y$ are complete (in fact, if $X$ and $Y$ are both infinite-dimensional Banach spaces then $X\; \backslash otimes\_\; Y$ is necessarily complete). However, $X\; \backslash otimes\_\; Y$ can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by $X\; \backslash widehat\_\; Y,$ via a linear topological embedding. Explicitly, this means that there is a continuous linear injective function, injection $\backslash operatorname\; :\; X\; \backslash otimes\_\; Y\; \backslash to\; X\; \backslash widehat\_\; Y$ whose image is dense in $X\; \backslash widehat\_\; Y$ and that is a TVS-isomorphism onto its image. Using this map, $X\; \backslash otimes\_\; Y$ is identified as a subspace of $X\; \backslash widehat\_\; Y.$ The continuous dual space of $X\; \backslash widehat\_\; Y$ is the same as that of $X\; \backslash otimes\_\; Y,$ namely the space of continuous bilinear forms $B(X,\; Y).$: Any continuous map on $X\; \backslash otimes\_\; Y$ can be extended to a unique continuous map on $X\; \backslash widehat\_\; Y.$ In particular, if $u\; :\; X\_1\; \backslash to\; Y\_1$ and $v\; :\; X\_2\; \backslash to\; Y\_2$ are continuous linear maps between locally convex spaces then their tensor product $u\; \backslash otimes\; v\; :\; X\_1\; \backslash otimes\_\; X\_2\; \backslash to\; Y\_1\; \backslash otimes\_\; Y\_2\; \backslash subseteq\; Y\_1\; \backslash widehat\_\; Y\_2,$ which is necessarily continuous, can be extended to a unique continuous linear function $u\; \backslash widehat\_\; v\; :\; X\_1\; \backslash widehat\_\; X\_2\; \backslash to\; Y\_1\; \backslash widehat\_\; Y\_2,$ which may also be denoted by $u\; \backslash widehat\; v$ if no ambiguity would arise. Note that if $X$ and $Y$ are metrizable then so are $X\; \backslash otimes\_\; Y$ and $X\; \backslash widehat\_\; Y,$ where in particular $X\; \backslash widehat\_\; Y$ will be an F-space.Grothendieck's representation of elements of $X\; \backslash widehat\_\; Y$

In a Hausdorff locally convex space $X,$ a sequence $\backslash left(x\_i\backslash right)\_^$ in $X$ is absolutely convergent if $\backslash sum\_^\; p\; \backslash left(x\_i\backslash right)\; <\; \backslash infty$ for every continuous seminorm $p$ on $X.$ We write $x\; =\; \backslash sum\_^\; x\_i$ if the sequence of partial sums $\backslash left(\backslash sum\_^n\; x\_i\backslash right)\_^$ converges to $x$ in $X.$ The following fundamental result in the theory of topological tensor products is due to Alexander Grothendieck. The next theorem shows that it is possible to make the representation of $z$ independent of the sequences $\backslash left(x\_i\backslash right)\_^$ and $\backslash left(y\_i\backslash right)\_^.$Topology of bi-bounded convergence

Let $\backslash mathfrak\_X$ and $\backslash mathfrak\_Y$ denote the families of all bounded subsets of $X$ and $Y,$ respectively. Since the continuous dual space of $X\; \backslash widehat\_\; Y$ is the space of continuous bilinear forms $B(X,\; Y),$ we can place on $B(X,\; Y)$ the topology of uniform convergence on sets in $\backslash mathfrak\_X\; \backslash times\; \backslash mathfrak\_Y,$ which is also called the topology of bi-bounded convergence. This topology is coarser than the strong topology $b\backslash left(B(X,\; Y),\; X\; \backslash widehat\_\; Y\backslash right),$ and in , Alexander Grothendieck was interested in when these two topologies were identical. This question is equivalent to the questions: Given a bounded subset $B\; \backslash subseteq\; X\; \backslash widehat\_\; Y,$ do there exist bounded subsets $B\_1\; \backslash subseteq\; X$ and $B\_2\; \backslash subseteq\; Y$ such that $B$ is a subset of the closed convex hull of $B\_1\; \backslash otimes\; B\_2\; :=\; \backslash $? Grothendieck proved that these topologies are equal when $X$ and $Y$ are both Banach spaces or both are DF-spaces (a class of spaces introduced by Grothendieck). They are also equal when both spaces are Fréchet with one of them being nuclear.Strong dual and bidual

Given a locally convex TVS $X,$ $X^$ is assumed to have the strong topology (so $X^\; =\; X^\_b$) and unless stated otherwise, the same is true of the bidual $X^$ (so $X^\; =\; \backslash left(X^\_b\backslash right)^\_b.$ Alexander Grothendieck characterized the strong dual and bidual for certain situations:Properties

* $X\; \backslash otimes\_\; Y$ is Hausdorff if and only if both $X$ and $Y$ are Hausdorff. * Suppose that $u\; :\; X\_1\; \backslash to\; Y\_1$ and $v\; :\; X\_2\; \backslash to\; Y\_2$ are two linear maps between locally convex spaces. If both $u$ and $v$ are continuous then so is their tensor product $u\; \backslash otimes\; v\; :\; X\_1\; \backslash otimes\_\; X\_2\; \backslash to\; Y\_1\; \backslash otimes\_\; Y\_2.$ ** $u\; \backslash otimes\; v\; :\; X\_1\; \backslash otimes\_\; X\_2\; \backslash to\; Y\_1\; \backslash otimes\_\; Y\_2$ has a unique continuous extension to $X\_1\; \backslash widehat\_\; X\_2$ denoted by $u\; \backslash widehat\; v\; :\; X\_1\; \backslash widehat\_\; X\_2\; \backslash to\; Y\_1\; \backslash widehat\_\; Y\_2.$ ** If in addition both $u$ and $v$ are TVS-homomorphisms and the image of each map is dense in its codomain, then $u\; \backslash widehat\_\; v\; :\; X\_1\; \backslash widehat\_\; X\_2\; \backslash to\; Y\_1\; \backslash widehat\_\; Y\_2$ is a homomorphism whose image is dense in $Y\_1\; \backslash widehat\_\; Y\_2$; if $X\_1$ and $Y\_1$ are both metrizable then this image is equal to all of $Y\_1\; \backslash widehat\_\; Y\_2.$ ** There are examples of $u$ and $v$ such that both $u$ and $v$ are surjective homomorphisms but $u\; \backslash widehat\_\; v\; :\; X\_1\; \backslash widehat\_\; X\_2\; \backslash to\; Y\_1\; \backslash widehat\_\; Y\_2$ is surjective. ** There are examples of $u$ and $v$ such that both $u$ and $v$ are TVS-embeddings but $u\; \backslash widehat\_\; v\; :\; X\_1\; \backslash widehat\_\; X\_2\; \backslash to\; Y\_1\; \backslash widehat\_\; Y\_2$ is a TVS-embedding. In order for $u\; \backslash widehat\_\; v$ to be a TVS-embedding, it is necessary and sufficient to additionally show that every equicontinuous subset of $B\backslash left(X\_1,\; X\_2\backslash right)$ is the image under $^\backslash left(u\; \backslash widehat\_\; v\backslash right)$ of an equicontinuous subset of $B\backslash left(Y\_1,\; Y\_2\backslash right).$ ** If all four spaces are normed then $\backslash ,\; u\; \backslash otimes\; v\; \backslash ,\; \_\; =\; \backslash ,\; u\; \backslash ,\; \backslash ,\; v\; \backslash ,\; .$ * The π-topology is finer than the Injective tensor product, ε-topology (since the canonical bilinear map $X\; \backslash times\; Y\; \backslash to\; X\; \backslash otimes\_\; Y$ is continuous). * If $X$ and $Y$ are Frechet spaces then $X\; \backslash otimes\_\; Y$ is barelled. * If $Y$ and $\backslash left(X\_\backslash right)$ are locally convex spaces then the canonical map $\backslash left(\backslash prod\_\; X\_\backslash right)\; \backslash widehat\_\; Y\; \backslash to\; \backslash prod\_\; \backslash left(X\_\; \backslash widehat\_\; Y\backslash right)$ is a TVS-isomorphism. * If $X$ and $Y$ are Frechet spaces and $Z$ is a complete Hausdorff locally convex space, then the canonical vector space isomorphism $I\; :\; B(X,\; Y;\; Z)\; \backslash to\; L\backslash left(X\; \backslash widehat\_\; Y;\; Z\backslash right)$ becomes a homeomorphism when these spaces are given the topologies of uniform convergence on products of compact sets and, for the second one, the topology of compact convergence (i.e. $I\; :\; B\_(X,\; Y;\; Z)\; \backslash to\; L\_\backslash left(X\; \backslash widehat\_\; Y;\; Z\backslash right)$ is a TVS-isomorphism). * Suppose $X$ and $Y$ are Frechet spaces. Every compact subset of $X\; \backslash widehat\_\; Y$ is contained in the closed convex balanced hull of the tensor product if a compact subset of $X$ and a compact subset of $Y.$ * If $X$ and $Y$ are nuclear then $X\; \backslash otimes\_\; Y$ and $X\; \backslash widehat\_\; Y$ are nuclear.Projective norm

Suppose now that $\backslash left(X,\; \backslash ,\; \backslash ,\backslash cdot\backslash ,\backslash ,\; \backslash right)$ and $\backslash left(Y,\; \backslash ,\; \backslash ,\backslash cdot\backslash ,\backslash ,\; \backslash right)$ are normed spaces. Then $X\; \backslash otimes\_\; Y$ is a normable space with a canonical norm denoted by $\backslash ,\; \backslash ,\backslash cdot\backslash ,\backslash ,\; \_.$ The $\backslash pi$-norm is defined on $X\; \backslash otimes\; Y$ by $$\backslash ,\; b\backslash ,\; \_\; :=\; \backslash inf\_\; r$$ where $W$ is the balanced convex hull of $C\_p\; \backslash otimes\; C\_q\; =\; \backslash left\backslash .$ Given $b$ in $X\; \backslash otimes\; Y,$ this can also be expressed as $$\backslash ,\; b\backslash ,\; \_\; :=\; \backslash inf\; \backslash sum\_i\; \backslash ,\; x\_i\; \backslash ,\; \backslash ,\; y\_i\; \backslash ,$$ where the infimum is taken over all finite sequences $x\_1,\; \backslash ldots,\; x\_n\; \backslash in\; X$ and $y\_1,\; \backslash ldots,\; y\_n\; \backslash in\; Y$ (of the same length) such that $b\; =\; x\_1\; \backslash otimes\; y\_1\; +\; \backslash cdots\; +\; x\_n\; \backslash otimes\; y\_n.$ If $b$ is in $X\; \backslash widehat\_\; Y$ then $$\backslash ,\; b\backslash ,\; \_\; :=\; \backslash inf\; \backslash sum\_i\; \backslash ,\; x\_i\; \backslash ,\; \backslash ,\; y\_i\; \backslash ,$$ where the infimum is taken over all (finite or infinite) sequences $x\_1,\; \backslash ldots,\; \backslash in\; X$ and $y\_1,\; \backslash ldots,\; \backslash in\; Y$ (of the same length) such that $b\; =\; x\_1\; \backslash otimes\; y\_1\; +\; \backslash cdots.$ Also, $$\backslash ,\; b\backslash ,\; \_\; :=\; \backslash inf\; \backslash sum\_i,\; \backslash lambda\_i,$$ where the infimum is taken over all sequences $\backslash left(x\_i\backslash right)$ in $X$ and $\backslash left(y\_i\backslash right)$ in $Y$ and scalars $\backslash lambda\_1,\; \backslash cdots$ (of the same length) such that $b\; =\; \backslash lambda\_1\; x\_1\; \backslash otimes\; y\_1\; +\; \backslash cdots,$ $\backslash ,\; x\_i\; \backslash ,\; =\; \backslash ,\; y\_i\; \backslash ,\; =\; 1,$ and $\backslash sum\_,\; \backslash lambda\_i,\; <\; \backslash infty.$ Also, $$\backslash ,\; b\backslash ,\; \_\; :=\; \backslash inf\; \backslash sum\_i,\; \backslash lambda\_i,\; \backslash ,\; x\_i\; \backslash ,\; \backslash ,\; y\_i\; \backslash ,$$ where the infimum is taken over all sequences $\backslash left(x\_i\backslash right)$ in $X$ and $\backslash left(y\_i\backslash right)$ in $Y$ and scalars $\backslash lambda\_1,\; \backslash cdots$ (of the same length) such that $b\; =\; \backslash lambda\_1\; x\_1\; \backslash otimes\; y\_1\; +\; \backslash cdots,$ $\backslash left(x\_i\backslash right)$ and $\backslash left(y\_i\backslash right)$ converge to the origin, and $\backslash sum\_,\; \backslash lambda\_i,\; <\; \backslash infty.$ If $X$ and $Y$ are Banach spaces then the closed unit ball of $X\; \backslash widehat\_\; Y$ is the closed convex hull of the tensor product of the closed unit ball in $X$ with that of $Y.$Properties

* For all normed spaces $\backslash left(Z,\; \backslash ,\; \backslash ,\backslash cdot\backslash ,\backslash ,\; \backslash right),$ the canonical vector space isomorphism of $B(X,\; Y;\; Z)$ onto $L\backslash left(X\; \backslash otimes\_\; Y;\; Z\backslash right)$ is an isometry. * Suppose that $\backslash ,\; \backslash ,\backslash cdot\backslash ,\backslash ,$ is a norm on $X\; \backslash otimes\; Y$ and let the TVS topology that it induces on $X\; \backslash otimes\; Y$ be denoted by $\backslash alpha.$ If the canonical linear map of $B(X,\; Y)$ into $\backslash left(X\; \backslash otimes\; Y\backslash right)^,$ which is the algebraic dual of $X\; \backslash otimes\; Y,$ is an isometry of $B(X,\; Y)$ onto $\backslash left(X\; \backslash otimes\_\; Y\backslash right)^,$ then $\backslash ,\; \backslash ,\backslash cdot\backslash ,\backslash ,\; =\; \backslash ,\; \backslash ,\backslash cdot\backslash ,\backslash ,\; \_.$Preserved properties

* In general, the projective tensor product does not respect subspaces (e.g. if $Z$ is a vector subspace of $X$ then the TVS $Z\; \backslash otimes\_\; Y$ has in general a coarser topology than the subspace topology inherited from $X\; \backslash otimes\_\; Y$). * Suppose that $E$ and $F$ are complemented subspaces of $X$ and $Y,$ respectively. Then $E\; \backslash otimes\; F$ is a complemented subvector space of $X\; \backslash otimes\_\; Y$ and the projective norm on $E\; \backslash otimes\_\; F$ is equivalent to the projective norm on $X\; \backslash otimes\_\; Y$ restricted to the subspace $E\; \backslash otimes\; F$; Furthermore, if $X$ and $F$ are complemented by projections of norm 1, then $E\; \backslash otimes\_\; F$ is complemented by a projection of norm 1. * If $I\; :\; X\; \backslash otimes\_\; Y\; \backslash to\; Z$ is an isometric embedding into a Banach space $Z,$ then its unique continuous extension $I\; :\; X\; \backslash widehat\_\; Y\; \backslash to\; Z$ is also an isometric embedding. * If $\backslash alpha\; :\; W\; \backslash to\; X$ and $\backslash beta\; :\; Y\; \backslash to\; Z$ are quotient operators between Banach spaces, then so is $\backslash alpha\; \backslash widehat\_\; \backslash beta\; :\; W\; \backslash widehat\_\; Y\; \backslash to\; X\; \backslash widehat\_\; Z.$ ** A continuous linear operator $\backslash beta\; :\; Y\; \backslash to\; Z$ between normed spaces is a quotient operator if it is surjective and it maps the open unit ball of $Y$ into the open unit ball of $Z,$ or equivalently if for all $z\; \backslash in\; Z,$ $\backslash ,\; z\; \backslash ,\; =\; \backslash inf\_\; \backslash ,\; y\backslash ,\; .$ * Let $X$ and $F$ be vector subspaces of the Banach spaces $X$ and $Y,$ respectively. Then $E\; \backslash widehat\_\; F$ is a TVS-subspace of $X\; \backslash widehat\_\; Y$ if and only if every bounded bilinear form on $E\; \backslash times\; F$ extends to a continuous bilinear form on $X\; \backslash times\; Y$ with the same norm.Trace form

Suppose that $X$ is a locally convex spaces. There is a bilinear form on $X\; \backslash times\; X^$ defined by $\backslash left(x,\; x^\backslash right)\; \backslash mapsto\; x^(x),$ which when $X$ is a Banach space has norm equal to 1. This bilinear form corresponds to a linear form on $X\; \backslash otimes\; X^$ given by mapping $z\; :=\; \backslash sum\_^n\; x\_i\; \backslash otimes\; x^\_$ to $\backslash sum\_^n\; x^\_i\backslash left(x\_i\backslash right)$ (where of course this value is in fact independent of the representation $\backslash sum\_^n\; x\_i\; \backslash otimes\; x^\_$ of $z$ chosen). Letting $X^$ have its strong dual topology, we can continuously extend this linear map to a map $\backslash operatorname\; :\; X\; \backslash widehat\_\; X^\_b\; \backslash to\; \backslash Complex$ (assuming that the vector spaces have scalar field $\backslash Complex$) called the trace of $X.$ This name originates from the fact that if we write $z\; =\; \backslash sum\_^n\; z\_\; e\_i\; \backslash otimes\; e\_j^$ where $e\_j^\backslash left(e\_i\backslash right)\; =\; 1$ if $i\; =\; j$ and 0 otherwise, then $\backslash operatorname(z)\; =\; \backslash sum\_^n\; z\_.$Duality with L(X; Y')

Assuming that $X$ and $Y$ are Banach spaces over the field $\backslash mathbb,$ one may define a dual system between $X\; \backslash widehat\_\; Y$ and $L\_b\backslash left(X;\; Y^\backslash right)$ with the duality map $\backslash left\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash right\backslash rangle\; :\; L\_b\backslash left(X;\; Y^\backslash right)\; \backslash times\; \backslash left(X\; \backslash widehat\_\; Y\backslash right)\; \backslash to\; \backslash mathbb$ defined by $\backslash langle\; u,\; z\; \backslash rangle\; :=\; \backslash operatorname\backslash left(\backslash left(u\; \backslash ,\; \backslash widehat\_\; \backslash operatorname\_Y\backslash right)\; (z)\backslash right),$ where $\backslash operatorname\_Y\; :\; Y\; \backslash to\; Y$ is the identity map and $u\; \backslash ,\; \backslash widehat\_\; \backslash operatorname\_Y\; :\; X\; \backslash widehat\_\; Y\; \backslash to\; Y^\; \backslash widehat\_\; Y$ is the unique continuous extension of the continuous map $u\; \backslash ,\; \backslash otimes\_\; \backslash operatorname\_Y\; :\; X\; \backslash otimes\_\; Y\; \backslash to\; Y^\; \backslash otimes\_\; Y.$ If we write $z\; =\; \backslash sum\_^\; \backslash lambda\_\; x\_i\; \backslash otimes\; y\_i$ with $\backslash sum\_^\; \backslash left,\; \backslash lambda\_i\; \backslash \; <\; \backslash infty$ and the sequences $\backslash left(x\_i\backslash right)\_^$ and $\backslash left(y\_i\backslash right)\_^$ each converging to zero, then we have$$\backslash left\backslash langle\; u,\; z\; \backslash right\backslash rangle\; =\; \backslash sum\_^\; \backslash lambda\_i\; \backslash left\backslash langle\; u\backslash left(x\_i\backslash right),\; y\_i\; \backslash right\backslash rangle.$$Nuclear operators

There is a canonical vector space embedding $I\; :\; X^\; \backslash otimes\; Y\; \backslash to\; L(X;\; Y)$ defined by sending $z\; :=\; \backslash sum\_^n\; x\_i^\; \backslash otimes\; y\_i$ to the map $$x\; \backslash mapsto\; \backslash sum\_i^n\; x\_i^(x)\; y\_i$$ where it can be shown that this value is independent of the representation of $z$ chosen.Nuclear operators between Banach spaces

Assuming that $X$ and $Y$ are Banach spaces, then the map $I\; :\; X^\_b\; \backslash otimes\_\; Y\; \backslash to\; L\_b(X;\; Y)$ has norm $1$ so it has a continuous extension to a map $\backslash hat\; :\; X^\_b\; \backslash widehat\_\; Y\; \backslash to\; L\_b(X;\; Y),$ where it is known that this map is not necessarily injective. The range of this map is denoted by $L^1(X;\; Y)$ and its elements are called nuclear operators. $L^1(X;\; Y)$ is TVS-isomorphic to $\backslash left(X^\_b\; \backslash widehat\_\; Y\backslash right)\; /\; \backslash operatorname\; \backslash hat$ and the norm on this quotient space, when transferred to elements of $L^1(X;\; Y)$ via the induced map $\backslash hat\; :\; \backslash left(X^\_b\; \backslash widehat\_\; Y\backslash right)\; /\; \backslash operatorname\; \backslash hat\; \backslash to\; L^1(X;\; Y),$ is called the trace-norm and is denoted by $\backslash ,\; \backslash ,\backslash cdot\backslash ,\backslash ,\; \_.$Nuclear operators between locally convex spaces

Suppose that $U$ is a convex balanced closed neighborhood of the origin in $X$ and $B$ is a convex balanced bounded Banach disk in $Y$ with both $X$ and $Y$ locally convex spaces. Let $p\_U(x)\; =\; \backslash inf\_\; r$ and let $\backslash pi\; :\; X\; \backslash to\; X/p\_U^(0)$ be the canonical projection. One can define the Auxiliary normed spaces, auxiliary Banach space $\backslash hat\_U$ with the canonical map $\backslash hat\_U\; :\; X\; \backslash to\; \backslash hat\_U$ whose image, $X/p\_U^(0),$ is dense in $\backslash hat\_U$ as well as the auxiliary space $F\_B\; =\; \backslash operatorname\; B$ normed by $p\_B(y)\; =\; \backslash inf\_\; r$ and with a canonical map $\backslash iota\; :\; F\_B\; \backslash to\; F$ being the (continuous) canonical injection. Given any continuous linear map $T\; :\; \backslash hat\_U\; \backslash to\; Y\_B$ one obtains through composition the continuous linear map $\backslash hat\_U\; \backslash circ\; T\; \backslash circ\; \backslash iota\; :\; X\; \backslash to\; Y$; thus we have an injection $L\; \backslash left(\backslash hat\_U;\; Y\_B\backslash right)\; \backslash to\; L(X;\; Y)$ and we henceforth use this map to identify $L\; \backslash left(\backslash hat\_U;\; Y\_B\backslash right)$ as a subspace of $L(X;\; Y).$ Let $X$ and $Y$ be Hausdorff locally convex spaces. The union of all $L^1\backslash left(\backslash hat\_U;\; Y\_B\backslash right)$ as $U$ ranges over all closed convex balanced neighborhoods of the origin in $X$ and $B$ ranges over all bounded Banach disks in $Y,$ is denoted by $L^1(X;\; Y)$ and its elements are call nuclear mappings of $X$ into $Y.$ When $X$ and $Y$ are Banach spaces, then this new definition of ''nuclear mapping'' is consistent with the original one given for the special case where $X$ and $Y$ are Banach spaces.Nuclear operators between Hilbert spaces

Every nuclear operator is an integral operator but the converse is not necessarily true. However, every integral operator between Hilbert spaces is nuclear.Nuclear bilinear forms

There is a canonical vector space embedding $J\; :\; X^\; \backslash otimes\; Y^\; \backslash to\; \backslash mathcal(X,\; Y)$ defined by sending $z\; :=\; \backslash sum\_^n\; x\_i^\; \backslash otimes\; y\_i^$ to the map $$(x,\; y)\; \backslash mapsto\; \backslash sum\_i^n\; x\_i^(x)\; y\_i(y)$$ where it can be shown that this value is independent of the representation of $z$ chosen.Nuclear bilinear forms on Banach spaces

Assuming that $X$ and $Y$ are Banach spaces, then the map $J\; :\; X^\_b\; \backslash otimes\_\; Y^\_b\; \backslash to\; \backslash mathcal\_b(X,\; Y)$ has norm $1$ so it has a continuous extension to a map $\backslash hat\; :\; X^\_b\; \backslash widehat\_\; Y^\_b\; \backslash to\; \backslash mathcal\_b(X,\; Y).$ The range of this map is denoted by $B^1(X,\; Y)$ and its elements are called nuclear bilinear forms. $B^1(X,\; Y)$ is TVS-isomorphic to $\backslash left(X^\_b\; \backslash widehat\_\; Y^\_b\backslash right)\; /\; \backslash operatorname\; \backslash hat$ and the norm on this quotient space, when transferred to elements of $B^1(X,\; Y)$ via the induced map $\backslash hat\; :\; \backslash left(X^\_b\; \backslash widehat\_\; Y^\_b\backslash right)\; /\; \backslash operatorname\; \backslash hat\; \backslash to\; B^1(X,\; Y),$ is called the nuclear-norm and is denoted by $\backslash ,\; \backslash ,\backslash cdot\backslash ,\backslash ,\; \_.$ Suppose that $X$ and $Y$ are Banach spaces and that $N$ is a continuous bilinear from on $X\; \backslash times\; Y.$ * The following are equivalent: # $N$ is nuclear. # There exist bounded sequences $\backslash left(x\_i^\backslash right)\_^$ in $X^\_b$ and $\backslash left(y\_i^\backslash right)\_^$ in $Y^\_b$ such that $\backslash sum\_^\; \backslash ,\; x\_i^\; \backslash ,\; \backslash ,\; y\_i^\; \backslash ,\; <\; \backslash infty$ and $N$ is equal to the mapping: $N(x,\; y)\; =\; \backslash sum\_^\; x^\_i(x)\; y^\_i(y)$ for all $(x,\; y)\; \backslash in\; X\; \backslash times\; Y.$ * In this case we call $\backslash sum\_^\; x^\_i\; \backslash otimes\; y^\_i$ a nuclear representation of $N.$ The nuclear norm of $N$ is: $$\backslash ,\; N\backslash ,\; \_\; =\; \backslash inf\; \backslash left\backslash .$$ Note that $\backslash ,\; N\backslash ,\; \backslash leq\; \backslash ,\; N\backslash ,\; \_.$Examples

Space of absolutely summable families

Throughout this section we fix some arbitrary (possibly uncountable) set $A,$ a TVS $X,$ and we let $\backslash mathcal(A)$ be the directed set of all finite subsets of $A$ directed by inclusion $\backslash subseteq.$ Let $\backslash left(x\_\backslash right)\_$ be a family of elements in a TVS $X$ and for every finite subset $H$ of $A,$ let $x\_H\; :=\; \backslash sum\_\; x\_i.$ We call $\backslash left(x\_\backslash right)\_$ summable in $X$ if the limit $\backslash lim\_\; x\_$ of the Net (mathematics), net $\backslash left(x\_H\backslash right)\_$ converges in $X$ to some element (any such element is called its sum). We call $\backslash left(x\_\backslash right)\_$ absolutely summable if it is summable and if for every continuous seminorm $p$ on $X,$ the family $\backslash left(p\; \backslash left(x\_\backslash right)\backslash right)\_$ is summable in $\backslash R.$ The set of all such absolutely summable families is a vector subspace of $X^$ denoted by $S\_a.$ Note that if $X$ is a metrizable locally convex space then at most countably many terms in an absolutely summable family are non-0. A metrizable locally convex space is nuclear space, nuclear if and only if every summable sequence is absolutely summable. It follows that a normable space in which every summable sequence is absolutely summable, is necessarily finite dimensional. We now define a topology on $S\_a$ in a very natural way. This topology turns out to be the projective topology taken from $\backslash ell^1(A)\; \backslash widehat\_\; X$ and transferred to $S\_a$ via a canonical vector space isomorphism (the obvious one). This is a common occurrence when studying the injective and projective tensor products of function/sequence spaces and TVSs: the "natural way" in which one would define (from scratch) a topology on such a tensor product is frequently equivalent to the projective or injective tensor product topology. Let $\backslash mathfrak$ denote a base of convex balanced neighborhoods of the origin in $X$ and for each $U\; \backslash in\; \backslash mathfrak,$ let $\backslash mu\_U\; :\; X\; \backslash to\; \backslash R$ denote its Minkowski functional. For any such $U$ and any $x\; =\; \backslash left(x\_\backslash right)\_\; \backslash in\; S\_a,$ let $p\_U(x)\; :=\; \backslash sum\_\; \backslash mu\_U\backslash left(x\_\backslash right)$ where $p\_U$ defines a seminorm on $S\_a.$ The family of seminorms $\backslash $ generates a topology making $S\_a$ into a locally convex space. The vector space $S\_a$ endowed with this topology will be denoted by $\backslash ell^1[A,\; X].$ The special case where $X$ is the scalar field will be denoted by $\backslash ell^1[A].$ There is a canonical embedding of vector spaces $\backslash ell^1(A)\; \backslash otimes\; X\; \backslash to\; \backslash ell^1[A,\; E]$ defined by linearizing the bilinear map $\backslash ell^1(A)\; \backslash times\; X\; \backslash to\; \backslash ell^1[A,\; E]$ defined by $\backslash left(\backslash left(r\_\backslash right)\_,\; x\backslash right)\; \backslash mapsto\; \backslash left(r\_\; x\backslash right)\_.$See also

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Bibliography

* * * * * * * * * * * * * * *External links

Nuclear space at ncatlab

{{Functional Analysis Functional analysis