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In functional analysis, an area of mathematics, the projective tensor product of two locally convex topological vector spaces is a natural topological vector space structure on their tensor product. Namely, given locally convex topological vector spaces X {\displaystyle X} and Y {\displaystyle Y} , the projective topology, or π-topology, on X ⊗ Y {\displaystyle X\otimes Y} is the strongest topology which makes X ⊗ Y {\displaystyle X\otimes Y} a locally convex topological vector space such that the canonical map ( x , y ) ↦ x ⊗ y {\displaystyle (x,y)\mapsto x\otimes y} (from X × Y {\displaystyle X\times Y} to X ⊗ Y {\displaystyle X\otimes Y} ) is continuous. When equipped with this topology, X ⊗ Y {\displaystyle X\otimes Y} is denoted X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} and called the projective tensor product of X {\displaystyle X} and Y {\displaystyle Y} . It is a particular instance of a topological tensor product.
Let X {\displaystyle X} and Y {\displaystyle Y} be locally convex topological vector spaces. Their projective tensor product X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} is the unique locally convex topological vector space with underlying vector space X ⊗ Y {\displaystyle X\otimes Y} having the following universal property:
When the topologies of X {\displaystyle X} and Y {\displaystyle Y} are induced by seminorms, the topology of X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} is induced by seminorms constructed from those on X {\displaystyle X} and Y {\displaystyle Y} as follows. If p {\displaystyle p} is a seminorm on X {\displaystyle X} , and q {\displaystyle q} is a seminorm on Y {\displaystyle Y} , define their tensor product p ⊗ q {\displaystyle p\otimes q} to be the seminorm on X ⊗ Y {\displaystyle X\otimes Y} given by ( p ⊗ q ) ( b ) = inf r > 0 , b ∈ r W r {\displaystyle (p\otimes q)(b)=\inf _{r>0,\,b\in rW}r} for all b {\displaystyle b} in X ⊗ Y {\displaystyle X\otimes Y} , where W {\displaystyle W} is the balanced convex hull of the set { x ⊗ y : p ( x ) ≤ 1 , q ( y ) ≤ 1 } {\displaystyle \left\{x\otimes y:p(x)\leq 1,q(y)\leq 1\right\}} . The projective topology on X ⊗ Y {\displaystyle X\otimes Y} is generated by the collection of such tensor products of the seminorms on X {\displaystyle X} and Y {\displaystyle Y} . When X {\displaystyle X} and Y {\displaystyle Y} are normed spaces, this definition applied to the norms on X {\displaystyle X} and Y {\displaystyle Y} gives a norm, called the projective norm, on X ⊗ Y {\displaystyle X\otimes Y} which generates the projective topology.
Throughout, all spaces are assumed to be locally convex. The symbol X ⊗ ^ π Y {\displaystyle X{\widehat {\otimes }}_{\pi }Y} denotes the completion of the projective tensor product of X {\displaystyle X} and Y {\displaystyle Y} .
In general, the space X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} is not complete, even if both X {\displaystyle X} and Y {\displaystyle Y} are complete (in fact, if X {\displaystyle X} and Y {\displaystyle Y} are both infinite-dimensional Banach spaces then X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} is necessarily not complete). However, X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by X ⊗ ^ π Y {\displaystyle X{\widehat {\otimes }}_{\pi }Y} .
The continuous dual space of X ⊗ ^ π Y {\displaystyle X{\widehat {\otimes }}_{\pi }Y} is the same as that of X ⊗ π Y {\displaystyle X\otimes _{\pi }Y} , namely, the space of continuous bilinear forms B ( X , Y ) {\displaystyle B(X,Y)} .
Grothendieck's representation of elements in the completion[edit]In a Hausdorff locally convex space X , {\displaystyle X,} a sequence ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} in X {\displaystyle X} is absolutely convergent if ∑ i = 1 ∞ p ( x i ) < ∞ {\displaystyle \sum _{i=1}^{\infty }p\left(x_{i}\right)<\infty } for every continuous seminorm p {\displaystyle p} on X . {\displaystyle X.} We write x = ∑ i = 1 ∞ x i {\displaystyle x=\sum _{i=1}^{\infty }x_{i}} if the sequence of partial sums ( ∑ i = 1 n x i ) n = 1 ∞ {\displaystyle \left(\sum _{i=1}^{n}x_{i}\right)_{n=1}^{\infty }} converges to x {\displaystyle x} in X . {\displaystyle X.}
The following fundamental result in the theory of topological tensor products is due to Alexander Grothendieck.
Theorem—Let X {\displaystyle X} and Y {\displaystyle Y} be metrizable locally convex TVSs and let z ∈ X ⊗ ^ π Y . {\displaystyle z\in X{\widehat {\otimes }}_{\pi }Y.} Then z {\displaystyle z} is the sum of an absolutely convergent series z = ∑ i = 1 ∞ λ i x i ⊗ y i {\displaystyle z=\sum _{i=1}^{\infty }\lambda _{i}x_{i}\otimes y_{i}} where ∑ i = 1 ∞ | λ i | < ∞ , {\displaystyle \sum _{i=1}^{\infty }|\lambda _{i}|<\infty ,} and ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} and ( y i ) i = 1 ∞ {\displaystyle \left(y_{i}\right)_{i=1}^{\infty }} are null sequences in X {\displaystyle X} and Y , {\displaystyle Y,} respectively.
The next theorem shows that it is possible to make the representation of z {\displaystyle z} independent of the sequences ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} and ( y i ) i = 1 ∞ . {\displaystyle \left(y_{i}\right)_{i=1}^{\infty }.}
Theorem—Let X {\displaystyle X} and Y {\displaystyle Y} be Fréchet spaces and let U {\displaystyle U} (resp. V {\displaystyle V} ) be a balanced open neighborhood of the origin in X {\displaystyle X} (resp. in Y {\displaystyle Y} ). Let K 0 {\displaystyle K_{0}} be a compact subset of the convex balanced hull of U ⊗ V := { u ⊗ v : u ∈ U , v ∈ V } . {\displaystyle U\otimes V:=\{u\otimes v:u\in U,v\in V\}.} There exists a compact subset K 1 {\displaystyle K_{1}} of the unit ball in ℓ 1 {\displaystyle \ell ^{1}} and sequences ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} and ( y i ) i = 1 ∞ {\displaystyle \left(y_{i}\right)_{i=1}^{\infty }} contained in U {\displaystyle U} and V , {\displaystyle V,} respectively, converging to the origin such that for every z ∈ K 0 {\displaystyle z\in K_{0}} there exists some ( λ i ) i = 1 ∞ ∈ K 1 {\displaystyle \left(\lambda _{i}\right)_{i=1}^{\infty }\in K_{1}} such that z = ∑ i = 1 ∞ λ i x i ⊗ y i . {\displaystyle z=\sum _{i=1}^{\infty }\lambda _{i}x_{i}\otimes y_{i}.}
Topology of bi-bounded convergence[edit]Let B X {\displaystyle {\mathfrak {B}}_{X}} and B Y {\displaystyle {\mathfrak {B}}_{Y}} denote the families of all bounded subsets of X {\displaystyle X} and Y , {\displaystyle Y,} respectively. Since the continuous dual space of X ⊗ ^ π Y {\displaystyle X{\widehat {\otimes }}_{\pi }Y} is the space of continuous bilinear forms B ( X , Y ) , {\displaystyle B(X,Y),} we can place on B ( X , Y ) {\displaystyle B(X,Y)} the topology of uniform convergence on sets in B X × B Y , {\displaystyle {\mathfrak {B}}_{X}\times {\mathfrak {B}}_{Y},} which is also called the topology of bi-bounded convergence. This topology is coarser than the strong topology on B ( X , Y ) {\displaystyle B(X,Y)} , and in (Grothendieck 1955), Alexander Grothendieck was interested in when these two topologies were identical. This is equivalent to the problem: Given a bounded subset B ⊆ X ⊗ ^ Y , {\displaystyle B\subseteq X{\widehat {\otimes }}Y,} do there exist bounded subsets B 1 ⊆ X {\displaystyle B_{1}\subseteq X} and B 2 ⊆ Y {\displaystyle B_{2}\subseteq Y} such that B {\displaystyle B} is a subset of the closed convex hull of B 1 ⊗ B 2 := { b 1 ⊗ b 2 : b 1 ∈ B 1 , b 2 ∈ B 2 } {\displaystyle B_{1}\otimes B_{2}:=\{b_{1}\otimes b_{2}:b_{1}\in B_{1},b_{2}\in B_{2}\}} ?
Grothendieck proved that these topologies are equal when X {\displaystyle X} and Y {\displaystyle 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[edit]Let X {\displaystyle X} be a locally convex topological vector space and let X ′ {\displaystyle X^{\prime }} be its continuous dual space. Alexander Grothendieck characterized the strong dual and bidual for certain situations:
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