In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space, a topological space, the latter thereby admitting a notion of continuity. More its topological space has a uniform topological structure, allowing a notion of uniform convergence; the elements of topological vector spaces are functions or linear operators acting on topological vector spaces, the topology is defined so as to capture a particular notion of convergence of sequences of functions. Hilbert spaces and Banach spaces are well-known examples. Unless stated otherwise, the underlying field of a topological vector space is assumed to be either the complex numbers C or the real numbers R. A topological vector space X is a vector space over a topological field K, endowed with a topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous functions; some authors require the topology on X to be T1. The topological and linear algebraic structures can be tied together more with additional assumptions, the most common of which are listed below.
The category of topological vector spaces over a given topological field K is denoted TVSK or TVectK. The objects are the topological vector spaces over K and the morphisms are the continuous K-linear maps from one object to another; every normed vector space has a natural topological structure: the norm induces a metric and the metric induces a topology. This is a topological vector space because: The vector addition +: V × V → V is jointly continuous with respect to this topology; this follows directly from the triangle inequality obeyed by the norm. The scalar multiplication ·: K × V → V, where K is the underlying scalar field of V, is jointly continuous; this follows from the triangle homogeneity of the norm. Therefore, all Banach spaces and Hilbert spaces are examples of topological vector spaces. There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis. Examples of such spaces are spaces of holomorphic functions on an open domain, spaces of infinitely differentiable functions, the Schwartz spaces, spaces of test functions and the spaces of distributions on them.
These are all examples of Montel spaces. An infinite-dimensional Montel space is never normable; the existence of a norm for a given topological vector space is characterized by Kolmogorov's normability criterion. A topological field is a topological vector space over each of its subfields. A cartesian product of a family of topological vector spaces, when endowed with the product topology, is a topological vector space. For instance, the set X of all functions f: R → R: this set X can be identified with the product space RR and carries a natural product topology. With this topology, X becomes a topological vector space, endowed with a topology called the topology of pointwise convergence; the reason for this name is the following: if is a sequence of elements in X fn has limit f in X if and only if fn has limit f for every real number x. This space is complete, but not normable: indeed, every neighborhood of 0 in the product topology contains lines, i.e. sets K f for f ≠ 0. A vector space is an abelian group with respect to the operation of addition, in a topological vector space the inverse operation is always continuous.
Hence, every topological vector space is an abelian topological group. Let X be a topological vector space. Given a subspace M ⊂ X, the quotient space X/M with the usual quotient topology is a Hausdorff topological vector space if and only if M is closed; this permits the following construction: given a topological vector space X, form the quotient space X / M where M is the closure of. X / M is a Hausdorff topological vector space that can be studied instead of X. In particular, topological vector spaces are uniform spaces and one can thus talk about completeness, uniform convergence and uniform continuity; the vector space operation of addition is uniformly continuous and the scalar multiplication is Cauchy continuous. Because of this, every topological vector space can be completed and is thus a dense linear subspace of a complete topological vector space; the Birkhoff–Kakutani theorem gives that the following three conditions on a topological vector space V are equivalent: The origin 0 is closed in V, there is a countable basis of neighborhoods for 0 in V.
V is metrizable. There is a translation-invariant metric on V that induces the given topology on V. A metric linear space means a vector space together with a metric for which addition and scalar multiplication are continuous. By the Birkhoff–Kakutani theorem, it follows that there is an equivalent metric, translation-invariant. More strongly: a topological vector space is said to be normable if its topology can be induced by a norm. A topological vector space is normable if and only if it is Hausdorff and has a convex bounded neighborhood of 0. A linear operator between two topological vector spaces, continuous at one point is continuous on the whole domain. Moreover, a linear operator f is continuous if f is bounded for some neighborhood V of 0. A hyperplane
Charles Roy Hauser was an American chemist. Hauser was a member of the National Academy of Sciences and a professor of chemistry at Duke University; the Sommelet–Hauser rearrangement is a named reaction based on the work of Hauser and Sommelet involving the rearrangement of certain benzyl quaternary ammonium salts. The reagent is sodium amide or another alkali metal amide and the reaction product a N,N-dialkylbenzylamine with a new alkyl group in the aromatic ortho position. For example, benzyltrimethylammonium iodide, I, rearranges in the presence of sodium amide to yield the o-methyl derivative of N,N-dimethylbenzylamine, his contributions were recognized by the following awards: 1957 the Florida Section Award 1962 the Herty Medal 1967 the Medal for Synthetic Organic Chemistry from the Synthetic Organic Chemical Manufacturers' Association
Kao Cheng-yan is an activist and founding chair of the Green Party Taiwan and a member of the Taiwan Environmental Protection Union. He was a Taiwan independence activist during his student years in the United States, he failed to gain a seat. In the 2004 ROC referendum, he debated DPP Legislator You Ching. In November 2019, Kao was ranked second on Green Party Taiwan's party list of legislative candidates contesting the 2020 elections, he opposes the completion of the Lungmen Nuclear Power Plant, leading the campaign to gather more than 120,000 signatures in order to add a referendum to the national ballot. His opposition to nuclear power dates back to 1979. Professionally, he is a professor in Computer Science at the National Taiwan University. 高成炎, ed.. 福島核災啟示錄：假如日本311發生在台灣……. Taipei: 前衛. ISBN 978-9578016842. "無標題文件". 2004. Archived from the original on June 11, 2004. Retrieved November 26, 2014. "Welcome to C. Y. Kao's Research Group!". Retrieved November 26, 2014