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In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis. One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology. ==The weak and strong topologies== Let ''K'' be a topological field, namely a field with a topology such that addition, multiplication, and division are continuous. In most applications ''K'' will be either the field of complex numbers or the field of real numbers with the familiar topologies. Let ''X'' be a topological vector space over ''K''. Namely, ''X'' is a ''K'' vector space equipped with a topology so that vector addition and scalar multiplication are continuous. We may define a possibly different topology on ''X'' using the continuous (or ''topological'') dual space ''X *''. The topological dual space consists of all linear functions from ''X'' into the base field ''K'' that are continuous with respect to the given topology. The weak topology on ''X'' is the initial topology with respect to ''X *''. In other words, it is the coarsest topology (the topology with the fewest open sets) such that each element of ''X *'' remains a continuous function. In order to distinguish the weak topology from the original topology on ''X'', the original topology is often called the strong topology. A subbase for the weak topology is the collection of sets of the form φ−1(''U'') where φ ∈ ''X'' * and ''U'' is an open subset of the base field ''K''. In other words, a subset of ''X'' is open in the weak topology if and only if it can be written as a union of (possibly infinitely many) sets, each of which is an intersection of finitely many sets of the form φ−1(''U''). More generally, if ''F'' is a subset of the algebraic dual space, then the initial topology of ''X'' with respect to ''F'', denoted by σ(''X'',''F''), is the weak topology with respect to ''F'' . If one takes ''F'' to be the whole continuous dual space of ''X'', then the weak topology with respect to ''F'' coincides with the weak topology defined above. If the field ''K'' has an absolute value , then the weak topology σ(''X'',''F'') is induced by the family of seminorms, : for all ''f''∈''F'' and ''x''∈''X''. In particular, weak topologies are locally convex. From this point of view, the weak topology is the coarsest polar topology; see weak topology (polar topology) for details. Specifically, if ''F'' is a vector space of linear functionals on ''X'' which separates points of ''X'', then the continuous dual of ''X'' with respect to the topology σ(''X'',''F'') is precisely equal to ''F'' . 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「Weak topology」の詳細全文を読む スポンサード リンク
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