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 and , the projective topology, or π-topology, on is the strongest topology which makes a locally convex topological vector space such that the canonical map (from to ) is continuous. When equipped with this topology, is denoted and called the projective tensor product of and .
Let and be locally convex topological vector spaces. Their projective tensor product is the unique locally convex topological vector space with underlying vector space having the following universal property: [1]
When the topologies of and are induced by seminorms, the topology of is induced by seminorms constructed from those on and as follows. If is a seminorm on , and is a seminorm on , define their tensor product to be the seminorm on given by
for all in , where is the balanced convex hull of the set . The projective topology on is generated by the collection of such tensor products of the seminorms on and . [2] [1] When and are normed spaces, this definition applied to the norms on and gives a norm, called the projective norm, on which generates the projective topology. [3]
Throughout, all spaces are assumed to be locally convex. The symbol denotes the completion of the projective tensor product of and .
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In general, the space is not complete, even if both and are complete (in fact, if and are both infinite-dimensional Banach spaces then is necessarily not complete [8] ). However, can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by .
The continuous dual space of is the same as that of , namely, the space of continuous bilinear forms . [9]
In a Hausdorff locally convex space a sequence in is absolutely convergent if for every continuous seminorm on [10] We write if the sequence of partial sums converges to in [10]
The following fundamental result in the theory of topological tensor products is due to Alexander Grothendieck. [11]
Theorem — Let and be metrizable locally convex TVSs and let Then is the sum of an absolutely convergent series
where and and are null sequences in and respectively.
The next theorem shows that it is possible to make the representation of independent of the sequences and
Theorem [12] — Let and be Fréchet spaces and let (resp. ) be a balanced open neighborhood of the origin in (resp. in ). Let be a compact subset of the convex balanced hull of There exists a compact subset of the unit ball in and sequences and contained in and respectively, converging to the origin such that for every there exists some such that
Let and denote the families of all bounded subsets of and respectively. Since the continuous dual space of is the space of continuous bilinear forms we can place on the topology of uniform convergence on sets in which is also called the topology of bi-bounded convergence. This topology is coarser than the strong topology on , 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 do there exist bounded subsets and such that is a subset of the closed convex hull of ?
Grothendieck proved that these topologies are equal when and are both Banach spaces or both are DF-spaces (a class of spaces introduced by Grothendieck [13] ). They are also equal when both spaces are Fréchet with one of them being nuclear. [9]
Let be a locally convex topological vector space and let be its continuous dual space. Alexander Grothendieck characterized the strong dual and bidual for certain situations:
Theorem [14] (Grothendieck) — Let and be locally convex topological vector spaces with nuclear. Assume that both and are Fréchet spaces, or else that they are both DF-spaces. Then, denoting strong dual spaces with a subscripted :
In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. A norm is a generalization of the intuitive notion of "length" in the physical world. If is a vector space over , where is a field equal to or to , then a norm on is a map , typically denoted by , satisfying the following four axioms:
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In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.
In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.
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In functional analysis and related areas of mathematics, a barrelled space is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them. Barrelled spaces were introduced by Bourbaki.
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.
In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite dimensional Euclidean spaces. They were introduced by Alexander Grothendieck.
In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products, but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle.
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In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces. One method is used if the disk is bounded: in this case, the auxiliary normed space is with norm
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs).
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In the mathematical discipline of functional analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domains is a subset of some finite-dimensional Euclidean space. It is possible to generalize the notion of derivative to functions whose domain and codomain are subsets of arbitrary topological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-valued function is a subset of a finite-dimensional Euclidean space then many of these notions become logically equivalent resulting in a much more limited number of generalizations of the derivative and additionally, differentiability is also more well-behaved compared to the general case. This article presents the theory of -times continuously differentiable functions on an open subset of Euclidean space , which is an important special case of differentiation between arbitrary TVSs. This importance stems partially from the fact that every finite-dimensional vector subspace of a Hausdorff topological vector space is TVS isomorphic to Euclidean space so that, for example, this special case can be applied to any function whose domain is an arbitrary Hausdorff TVS by restricting it to finite-dimensional vector subspaces.
This is a glossary for the terminology in a mathematical field of functional analysis.
In functional analysis and related areas of mathematics, a metrizable topological vector space (TVS) is a TVS whose topology is induced by a metric. An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.
In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued functions on a non-empty open subset that have compact support. The space of all test functions, denoted by is endowed with a certain topology, called the canonical LF-topology, that makes into a complete Hausdorff locally convex TVS. The strong dual space of is called the space of distributions on and is denoted by where the "" subscript indicates that the continuous dual space of denoted by is endowed with the strong dual topology.