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Injective Tensor Product
In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the . Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y with any need to extend definitions (such as "differentiable at a point") from real/complex-valued functions to Y-valued functions. Preliminaries and notation Throughout let X, Y, and Z be 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Vector Space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also Continuous function, continuous functions. Such a topology is called a and every topological vector space has a Uniform space, uniform topological structure, allowing a notion of uniform convergence and Complete topological vector space, completeness. Some authors also require that the space is a Hausdorff space (although this article does not). One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs. Many topological vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barrelled Space
In functional analysis and related areas of mathematics, a barrelled space (also written barreled 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 . Barrels A convex and balanced subset of a real or complex vector space is called a and it is said to be , , or . A or a in a topological vector space (TVS) is a subset that is a closed absorbing disk; that is, a barrel is a convex, balanced, closed, and absorbing subset. Every barrel must contain the origin. If \dim X \geq 2 and if S is any subset of X, then S is a convex, balanced, and absorbing set of X if and only if this is all true of S \cap Y in Y for every 2-dimensional vector subspace Y; thus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. In the case of a logical matrix representing a binary relation R, the transpose corresponds to the converse relation RT. Transpose of a matrix Definition The transpose of a matrix , denoted by , , , A^, , , or , may be constructed by any one of the following methods: # Reflect over its main diagonal (which runs from top-left to bottom-right) to obtain #Write the rows of as the columns of #Write the columns of as the rows of Formally, the -th row, -th column element of is the -th row, -th column element of : :\left mathbf^\operatorname\right = \left mathbf\right. If is an matrix, then is an matrix. In the case of square matrices, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nuclear Space
In mathematics, nuclear spaces are topological vector space, 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. The topology on nuclear spaces can be defined by a family of seminorms whose Unit sphere#Unit balls in normed vector spaces, unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold. All finite-dimensional vector spaces are nuclear. There are no Banach space, Banach spaces that are nuclear, except for the finite-dimensional ones. In practice a sort of converse to this is often true: if a "naturally occurring" topological vector space is a Banach space, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nuclear Operator
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). Preliminaries and notation Throughout let ''X'',''Y'', and ''Z'' be topological vector spaces (TVSs) and ''L'' : ''X'' → ''Y'' be a linear operator (no assumption of continuity is made unless otherwise stated). * The projective tensor product of two locally convex TVSs ''X'' and ''Y'' is denoted by X \otimes_ Y and the completion of this space will be denoted by X \widehat_ Y. * ''L'' : ''X'' → ''Y'' is a topological homomorphism or homomorphism, if it is linear, continuous, and L : X \to \operatorname L is an open map, where \operatorname L, the image of ''L'', has the subspace topology induced by ''Y''. ** If ''S'' is a subspace of ''X'' then both the quotient map ''X'' → ''X''/''S'' and the canonical injection ''S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Tensor Product
The strongest 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 projective topology or the π-topology. When X \otimes Y is endowed with this topology then it is denoted by X \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 \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 L is an open map, where \operatorname 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 f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Dual Space
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of X, where this topology is denoted by b\left(X^, X\right) or \beta\left(X^, X\right). The coarsest polar topology is called weak topology. The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise. To emphasize that the continuous dual space, X^, has the strong dual topology, X^_b or X^_ may be written. Strong dual topology Throughout, all vector spaces will be assumed to be over the field \mathbb of either the real numbers \R or complex numbers \C. Definition from a dual system Let (X, Y, \langle \cdot, \cdot \rangle) be a dual pair of vector spaces over the field \mathbb of real numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparison Of Topologies
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the collection of subsets which are considered to be "open". An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following, it doesn't matter which definition is used. Let ''τ''1 and ''τ''2 be two topologies on a set ''X'' such that ''τ''1 is contained in ''τ''2: :\tau_1 \subseteq \tau_2. That is, every element of ''τ''1 is also an element of ''τ''2. Then the topology ''τ''1 is said to be a coarser (weaker or smaller) topology than ''τ''2, and ''τ''2 is said to be a finer (stronger or larger) topology than ''τ''1. There are some authors, especially ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. "Strongest" is Wroldsen's first solo release on Syco Music Syco Music is a defunct division of Syco Entertainment founded by British entrepreneur and record executive Simon Cowell. Originally founded as S Records, the label launched while Cowell was still employed by BMG, the label oversaw music relea ... after signing to the label in June 2017. The track centers its message on the heartbreak of a lost love and moving forward with her son. In an interview with Charlotte Sissener of ''Musikknyheter'', Wroldsen said the song is about a good friend of hers, saying the situation is "hard and bitter, but also genuine and triumphant." Track listing Charts Weekly charts Year-end charts Release history References {{Alan Walker 2017 songs 2017 singles Ina Wroldsen songs Songs written by Arnthor Birgisson ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
TVS-embedding
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space (although this article does not). One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs. Many topological vector spaces are spaces of functions, or linear operators acting on top ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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_ 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 L is an open map, where \operatorname 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 canonicall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
