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# Garrett P.'s Functions on circles PDF

By Garrett P.

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Rn below rn+1 . Then use the first observation of this argument to find Vrn+1 such that Vrj ⊂ V rj ⊂ Vrn+1 ⊂ V rn+1 Vri ⊂ V ri This constructs the nested family of opens. Let f (x) be the sup and g(x) the inf of the characteristic functions above. If f (x) > g(x) then there are r > s such that x ∈ Vr and x ∈ V s . But r > s implies that Vr ⊂ V s , so this cannot happen. If g(x) > f (x), then there are rationals r > s such that g(x) > r > s > f (x) Then s > f (x) implies that x ∈ Vs , and r < g(x) implies x ∈ V r .

13. Appendix: Frechet spaces and limits of Banach spaces A somewhat larger class of topological vector spaces that arise very often in practice is the class of Fr´ echet spaces. In our present context, we can give a nice definition: a Fr´echet space is a countable limit of Banach spaces. ☎ ✝ Thus, for example, C∞ (S 1 ) = Ck (S 1 ) = lim Ck (S 1 ) k k is a Fr´echet space, by definition. The present definition, whatever its advantages, is not the usual definition. elaborate somewhat on the features of Fr´echet spaces.

W xn ⊂ U Thus, the open set V = W ∩ W x1 ∩ . . ∩ W xn 39 Paul Garrett: Functions on circles (April 21, 2006) meets the requirements. Using the possibility of inserting an open subset and its closure between any K ⊂ U with K compact and U open, we will inductively create opens Vr (with compact closures) indexed by rational numbers r in the interval 0 ≤ r ≤ 1 such that, for r > s, we have the relation K ⊂ Vr ⊂ V r ⊂ Vs ⊂ V s ⊂ U From any such configuration of opens we will construct the desired sort of continuous function f by f (x) = sup{r rational in [0, 1] : x ∈ Vr , } = inf{r rational in [0, 1] : x ∈ V r , } It is not completely immediate that this sup and inf are the same, but if we grant their equality then we can prove the continuity of this function f (x).