拓撲向量空間

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拓撲向量空間是泛函分析研究中其中一個基本結構。顧名思義就是要研究具有拓撲結構的向量空間。

拓撲向量空間主要都是函數空間，在上面定義的拓撲結構就是函數列收歛的條件。

希爾伯特空間及巴那赫空間是典型的例子。

目录 1 定義 2 實例 2.1 乘積向量空間 3 拓撲結構 4 拓撲向量空間的種類 5 雙對空間 6 參考

[编辑] 定義

拓撲向量空間 X是實向量空間或複向量空間。其拓撲空間內，向量加法 (X × X → X) 及標量乘法(K × X → X) 都是連續性函數。

有時侯, 會要求拓撲向量空間上的拓撲結構是豪斯多夫空間:Hausdorff的。

[编辑] 實例

     所有線性賦範向量空間 (希爾伯特空間及巴那赫空間)都是例子。

[编辑] 乘積向量空間

數個 (不一定有限個)拓撲向量空間的笛卡爾積是拓撲向量空間，上面的拓撲結構是一般的乘積拓撲。

例如：X 是所有函數f的集合: R → R。 X 等於R^R 加上乘積拓撲。透過這個拓撲，X變成的拓撲向量空間稱為點式匯合空間。

獲得這個名稱的原因是: 如果 (f_n) 是X的函數列,那麼 f_n 以 f 為極限的充要條件是 對所有實數x中f_n(x) 有個極限f(x) 。 這是一個完備的空間，但它的拓撲不可能用範(norm)來定義。

[编辑] 拓撲結構

向量空間是有加法的阿貝爾群，其逆運算是連續的。因此每個拓撲向量空間都是阿貝爾拓撲群。

拓撲向量空間是均勻空間 以及可以讨论完备性, 均勻聚度 和 均勻連續性. 向量空间的加法和数乘运算实际上是一致连续的. 正因如此, 每个拓扑向量空间可以完备化并且是一个完备拓扑向量空间的一个稠密线性子空间.

在一個向量拓撲空間內的加法和向量乘法不僅是連續的,它們可以成為從這個空間到自己的同胚映射: 因為給定零點附近的local base, 我可以構作整個拓撲.

每一個拓撲向量空間都存在一組absorbing和 balanced的local base.

如果一個拓撲向量空間是 semi-metrisable的 (在其上可以定義semi-metric), 它上面的 semi-metric 必定是 translation invariant.

當在兩個向量空間之間的一個線性函數是連續於一點,便是連續於整個域。

在一個拓撲向量空間"X"上的線性泛函是連續的充要條件是: f 的核 是X中的閉集.

A topological vector space is finite-dimensional if and only if it is locally compact, in which case it is isomorphic to a Euclidean space Rⁿ or Cⁿ (in the sense that there exists a linear homeomorphism between the two spaces).

[编辑] 拓撲向量空間的種類

Depending on the application we usually enforce additional constraints on the topological structure of the space. Below are some common topological vector spaces, roughly ordered by their niceness.

• Locally convex topological vector spaces: here each point has a local base consisting of convex sets. By a technique known as Minkowski functionals it can be shown that a space is locally convex if and only if its topology can be defined by a family of semi-norms. Local convexity is the minimum requirement for "geometrical" arguments.
• Barrelled spaces: locally convex spaces where the Banach-Steinhaus theorem holds.
• Montel 空间: 如果桶空间中每个闭的 同时为 有界的集 是 紧的
• Bornological space: a locally convex space where the continuous linear operators to any locally convex space are exactly the bounded linear operators.
• F-spaces are complete 拓撲向量空間 with a translation-invariant metric. These include L^p spaces for all p > 0.
• Fréchet spaces: these are complete locally convex spaces where the topology comes from a translation-invariant metric, or equivalently: from a countable family of semi-norms. Many interesting spaces of functions fall into this class. A locally convex F-space is a Fréchet space.
• 線性賦範空間及半線性賦範空間: locally convex spaces where the topology can be described by a single 範數 or 半線性賦範. In normed spaces a linear operator is continuous if and only if it is bounded.
• 巴那赫空間: 完整線性賦範向量空間. 確立大部分函數分析.
• Reflexive Banach spaces: Banach spaces naturally isomorphic to their double dual (see below), which ensures that some geometrical arguments can be carried out. An important example which is not reflexive is L¹, whose dual is L^∞ but is strictly contained in the dual of L^∞.
• 希爾伯特空間: 它有點積; 縱使有無限維, 大部分幾何推理跟有限維空間是一樣的。

[编辑] 雙對空間

Every topological vector space has a 連續雙對空間—the set V^* of all continuous linear functionals, i.e. continuous linear maps from the space into the base field K. A topology on the dual can be defined to be the coarsest topology such that the dual pairing V^* × V → K is continuous. This turns the dual into a locally convex topological vector space. This topology is called the weak-* topology. This may not be the only natural topology on the dual space; for instance, the dual of a Banach space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see Banach-Alaoglu theorem).

[编辑] 參考

• A Grothendieck: Topological vector spaces, Gordon and Breach Science Publishers, New York, 1973.
• G Köthe: Topological vector spaces. Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag, New York, 1969.
• H H Schaefer: Topological vector spaces, Springer-Verlag, New York, 1971.
• F Trèves: Topological Vector Spaces, Distributions, and Kernels, Academic Press, 1967.