|
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus ''sequences'' of functions. The equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of ''C''(''X''), the space of continuous functions on a compact Hausdorff space ''X'', is compact if and only if it is closed, pointwise bounded and equicontinuous. As a corollary, a sequence in ''C''(''X'') is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions ''fn'' on either metric space or locally compact space〔More generally, on any compactly generated space; e.g., a first-countable space.〕 is continuous. If, in addition, ''fn'' are holomorphic, then the limit is also holomorphic. The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous. ==Definition== Let ''X'' and ''Y'' be two metric spaces, and ''F'' a family of functions from ''X'' to ''Y''. The family ''F'' is equicontinuous at a point ''x''0 ∈ ''X'' if for every ε > 0, there exists a δ > 0 such that ''d''(''ƒ''(''x''0), ''ƒ''(''x'')) < ε for all ''ƒ'' ∈ ''F'' and all ''x'' such that ''d''(''x''0, ''x'') < δ. The family is equicontinuous if it is equicontinuous at each point of ''X''.〔, p. 29; , p. 245〕 The family ''F'' is uniformly equicontinuous if for every ε > 0, there exists a δ > 0 such that ''d''(''ƒ''(''x''1), ''ƒ''(''x''2)) < ε for all ''ƒ'' ∈ ''F'' and all ''x''1, ''x''2 ∈ ''X'' such that ''d''(''x''1, ''x''2) < δ.〔, p. 29〕 For comparison, the statement 'all functions ''ƒ'' in ''F'' are continuous' means that for every ε > 0, every ''ƒ'' ∈ ''F'', and every ''x''0 ∈ ''X'', there exists a δ > 0 such that ''d''(''ƒ''(''x''0), ''ƒ''(''x'')) < ε for all ''x'' ∈ ''X'' such that ''d''(''x''0, ''x'') < δ. * For ''continuity'', δ may depend on ε, ''x''0 and ''ƒ''. * For ''uniform continuity'', δ may depend on ε, and ''ƒ''. * For ''equicontinuity'', δ may depend on ε, and ''x''0. * For ''uniform equicontinuity'', δ may solely depend on ε. More generally, when ''X'' is a topological space, a set ''F'' of functions from ''X'' to ''Y'' is said to be equicontinuous at ''x'' if for every ε > 0, ''x'' has a neighborhood ''Ux'' such that : for all and ''ƒ'' ∈ ''F''. This definition usually appears in the context of topological vector spaces. When ''X'' is compact, a set is uniformly equicontinuous if and only if it is equicontinuous at every point, for essentially the same reason as that uniform continuity and continuity coincide on compact spaces. Some basic properties follow immediately from the definition. Every finite set of continuous functions is equicontinuous. The closure of an equicontinuous set is again equicontinuous. Every member of a uniformly equicontinuous set of functions is uniformly continuous, and every finite set of uniformly continuous functions is uniformly equicontinuous. 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「Equicontinuity」の詳細全文を読む スポンサード リンク
|