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Projective Tensor Product
The 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 after signing to the label in ... locally convex 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 (an Abstra ... (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 Through ...
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Strong Topology
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a strong topology is a topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to: * the final topology In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a Set (mathematics), set X, with respect to a family of functions from Topological space, topological spaces into X, ... on the disjoint uni ...
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Normable Space
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a normed vector space or normed space is a vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... over the real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ... or complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , motto ...
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Banach Disk
In functional analysis, two methods of constructing Normed vector space, normed spaces from Absolutely convex set, disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces. One method is used if the disk D is bounded: in this case, the auxiliary normed space is \operatorname D with norm p_D(x) := \inf_ r. The other method is used if the disk D is absorbing set, absorbing: in this case, the auxiliary normed space is the Quotient space (linear algebra), quotient space X / p_D^(0). If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic (as topological vector spaces and as normed spaces). Preliminaries A subset of a vector space is called a Absolutely convex set, disk and is said to be disked, Absolutely convex set, absolutely convex, or convex balanced if it is Convex set, convex and balanced set, balanced. If C and D are subsets of a vector space X then D Absorbing set, absorb ...
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Dual System
In the field of functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ..., a subfield of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a dual system, dual pair, or a duality over a field \mathbb (\mathbb is either the real or the complex numbers) is a triple (X, Y, b) consisting of two vector spaces over \mathbb and a bilinear map b : X \times Y \to \mathbb such that for all non-zero x \in X the map y \mapsto b(x, y) is not identically 0 and for all non-zero y \in Y, the map x \mapsto b(x, y) is not identically 0. The study of dual systems is called dua ...
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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 topological vector space, 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 topological vector space, locally convex TVS with any need to extend definitions (such as "differentiable at a point") from real/complex-valued functions to -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 t ...
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Reflexive Space
In the area of mathematics known as functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ..., a reflexive space is a locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological spa ... 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 (an Abstra ... (TVS) such that the canonical evaluation map from X into its bidual (which is the strong dual In functional analysi ...
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DF-space
In the field of functional analysis, DF-spaces, also written (''DF'')-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the theory of topological tensor products. DF-spaces were first defined by Alexander Grothendieck and studied in detail by him in . Grothendieck was led to introduce these spaces by the following property of strong duals of metrizable spaces: If X is a Metrizable topological vector space, metrizable locally convex space and V_1, V_2, \ldots is a sequence of convex 0-neighborhoods in X^_b such that V := \cap_ V_i absorbs every strongly bounded set, then V is a 0-neighborhood in X^_b (where X^_b is the continuous dual space of X endowed with the strong dual topology). Definition A locally convex topological vector space (TVS) X is a DF-space, also written (''DF'')-space, if # X is a countably quasi-barrelled space (i.e. every strongly bou ...
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Fréchet Space
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 (normed vector spaces that are Complete space, complete with respect to the Metric (mathematics), metric induced by the Norm (mathematics), norm). All Banach space, Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable Function (mathematics), functions are typical examples of Fréchet spaces, many of which are typically Banach spaces. A Fréchet space X is defined to be a Locally convex topological vector space, locally convex Metrizable topological vector space, metrizable topological vector space (TVS) that is Complete topological vector space, complete as a TVS, meaning that every Cauchy sequence in X converges to some point in X (see footnote for more details).Here "Cauchy" means Cauchy with respect to the Complete topological vector space, canonical uniformity ...
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Null Sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the \lim symbol (e.g., \lim_a_n).Courant (1961), p. 29. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests. Limits can be defined in any metric space, metric or topological space, but are usually first encountered in the real numbers. History The Greek philosopher Zeno of Elea is famous for formulating Zeno's paradoxes, paradoxes that involve limiting processes. Leucippus, Democritus, Antiphon (person), Antiphon, Eudoxus of Cnidus, Eudoxus, and Archimedes deve ...
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Absolutely Convergent
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., an infinite series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...s of the summands is finite. More precisely, a real Real may refer to: * Reality Reality ...
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Alexander Grothendieck
Alexander Grothendieck (; ; ; 28 March 1928 – 13 November 2014) was a mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory and category theory to its foundations, while his so-called Grothendieck's relative point of view, "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the 20th century. Born in Germany, Grothendieck was raised and lived primarily in France, and he and his family were persecuted by the Nazi Germany, Nazi regime. For much of his working life, however, he was, in effect, Statelessness, stateless. As he consistently spelled his first name "Alexander" rather than "Alexandre" and his surname, taken from his mother, was the Dutch-like Low German "Grothendieck", he was sometimes mistakenly believed to be of D ...
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F-space
In functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ..., an F-space is a vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... X over the real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ... or complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come w ...
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