# Locally Convex Spaces by Kelly McKennon

By Kelly McKennon

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Vi) implies F' = {f^o 3: 0 € (B ^ n^Z (viii) Since bears F*-"b . S' = a(F, F ’) . xi) below. x) . sequence {s(n)} (V п Ш ) b^F in To this end, we first show N (V f^<»F€D: and a sequence f 0, inductively as follows: if {o } let and, if such that • k=l (V j€N) PO 4 | £ ^( n) o3 ( b) | = 12 < ^^ . x) is false. fjj(n) OÔÇD, is defined; F ’, then + = 0 . and suppose that (3 j ^N) s(j) = n , 2

Xv). 3. The Locally Convex Space F = ГТ+ (i) n 6Z Elements of Q b F F . Let (n) ] [+ n^Z will be represented by lower case German script: C ....... *-“ o ^ (iii) 3 (0 )^ = + pA(n) (Oj^ j) (V n€N). f 0, then non-zero. n(Z ¢ 0 . implies that F into ] n^Z for which If 0 . n = 0, then S(o)^ = 'f'FC^(“ n^ + Hence, 3 (q ) Ф O . ^ Then ^( û ) q =Yp^^(o) Ф 0. iv) holds is now evident. Let (v) S' be the coarsest topology on ] I^ n^Z Thus, if F (vi) For each 3 is continuous S' is a topological isomorphism.

9], (iii) E’ is (E', E ) -Montel . 28 For each n€N (iv) and = {XŒ: r > 0 , let ||x||<^'^^ £ Г} . Then the family : n^N, r > 0 } (V) system of constitutes a base for the 0 -neighborhood E The polars of the 0-neighborhoods of E are just the closed, absolutely convex, equicontinuous subsets of . S]. 1. 6 ]. continuous precisely the (2. 2], it follows from that the subsets. (n)‘ be defined by :(n)’. ) . E' = . Since (V n€N) a function in пШ r>0 Дп)^ is in Дп) E'^“ ^ if it is bounded on B^^^ сг)(Е^^^ ) (V r > (viii) Let E' = 5' when dently 0).