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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Topology of uniform convergence ： ウィキペディア英語版 Topology of uniform convergence In mathematics, a linear map is a mapping between two modules (including vector spaces) that preserves the operations of addition and scalar multiplication. By studying the linear maps between two modules one can gain insight into their structures. If the modules have additional structure, like topologies or bornologies, then one can study the subspace of linear maps that preserve this structure. ==Topologies of uniform convergence== Suppose that ''T'' be any set and that $\backslash mathcal$ be collection of subsets of ''T'' directed by inclusion. Suppose in addition that ''Y'' is a topological vector space (not necessarily Hausdorff or locally convex) and that $\backslash mathcal$ is a basis of neighborhoods of 0 in ''Y''. Then the set of all functions from ''T'' into ''Y'', $Y^T$, can be given a unique translation-invariant topology by defining a basis of neighborhoods of 0 in $Y^T$, to be :$\backslash mathcal(G,\; N)\; =\; \backslash$ as ''G'' and ''N'' range over all $G\; \backslash in\; \backslash mathcal$ and $N\; \backslash in\; \backslash mathcal$. This topology does not depend on the basis $\backslash mathcal$ that was chosen and it is known as the topology of uniform convergence on the sets in $\backslash mathcal$ or as the $\backslash mathcal$-topology.〔Schaefer (1970) p. 79〕 In practice, $\backslash mathcal$ usually consists of a collection of sets with certain properties and this name is changed appropriately to reflect this set so that if, for instance, $\backslash mathcal$ is the collection of compact subsets of ''T'' (and ''T'' is a topological space), then this topology is called the topology of uniform convergence on the compact subsets of ''T''. A set $\backslash mathcal\_1$ of $\backslash mathcal$ is said to be fundamental with respect to $\backslash mathcal$ if each $G\; \backslash in\; \backslash mathcal$ is a subset of some element in $\backslash mathcal\_1$. In this case, the collection $\backslash mathcal$ can be replaced by $\backslash mathcal\_1$ without changing the topology on $Y^T$.〔 However, the $\backslash mathcal$-topology on $Y^T$ is not necessarily compatible with the vector space structure of $Y^T$ or of any of its vector subspaces (that is, it is not necessarily a topological vector space topology on $Y^T$). Suppose that ''F'' is a vector subspace $Y^T$ so that it inherits the subspace topology from $Y^T$. Then the $\backslash mathcal$-topology on ''F'' is compatible with the vector space structure of ''F'' if and only if for every $G\; \backslash in\; \backslash mathcal$ and every ''f'' ∈ ''F'', ''f''(''G'') is bounded in ''Y''.〔 If ''Y'' is locally convex then so is the $\backslash mathcal$-topology on $Y^T$ and if $(p\_)$ is a family of continuous seminorms generating this topology on ''Y'' then the $\backslash mathcal$-topology is induced by the following family of seminorms: $p\_(f)\; =\; \backslash sup\_\; p\_(f(x))$, as ''G'' varies over $\backslash mathcal$ and $\backslash alpha$ varies over all indices.〔Schaefer (1970) p. 81〕 If ''Y'' is Hausdorff and ''T'' is a topological space such that $\backslash bigcup\_$-topology on subspace of $Y^T$ consisting of all continuous maps is Hausdorff. If the topological space ''T'' is also a topological vector space, then the condition that $\backslash bigcup\_$-topology if and only if for every $G\; \backslash in\; \backslash mathcal$, $\backslash cup\_\; u(G)$ is bounded in ''Y''.〔Schaefer (1970) p. 81〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Topology of uniform convergence」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.