Banach Spaces by Jeffrey M. Lemm

By Jeffrey M. Lemm

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Sample text

Thus II~(s) - y(s)ll < II~(s) - x(t)ll + Ilk(t) - ~ ( ~ d l l + c + II~(s,) - y ( s , ) l I + Ily(s,) - y(t)ll + Ily(t) - y(~)ll < 5 g = c. T" of pointwise convergence in A is finer than the topology on ~ of uniform convergence on T . Since the reverse relation is trivial, the two topologies on ~ coincide. 16 ( 0 ) I (Ascoli, 1883; Arzels 1889) Let T be a compact space. A set of C(T) is relatively compact iff it is bounded and equicontinuous. First assume t h a t ~ is a relatively compact set of C(T).

EJ Now assume that E is complete and that the condition described above is fulfilled.

Then there is a finite subset G of C(T) with yE6 Furthermore, there is a neighbourhood U of t with c ly(~)- y(t)l < 5 whenever y C G and s C U. Take x C ~ and s C U. Then there is a y C G with I I x - yll < ~. Thus c c c I~(~) - z(t)l _< Ix(~) - y(~)l + ly(~) - y(t)l + ly(t) - ~(t)l < 5 + 5 + 5 - ~' 24 1. Banach Spaces and so 9v is equicontinuous. ~" is bounded and equicontinuous. Take an ultrafilter on ~ . Define x-T ~IK t, ' ~limy(t). y,~ Choose t E T and r > 0. There is a neighbourhood U of t with ly(~)- y(t)l < whenever y E $" and s E U .

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