By P. Hoffman, R. Piccinini, D. Sjerve
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Other characterizations of locally decomposable and locally solid spaces can be fou2~l in Wong and Ng [I], while characterizations of locally o-convex spaces c~u be found in Schaefer [i], Peressini [I] and Jameson . s(v) . v ~ It is e~;~ily seen that denoted by be an ordered convex space and ~ F(~) ~i Setting . is the least upper bound of all locally o-convex topologies which arecoarser than2 . This topology locally o-convex %p~ologv associated with 2 . S(~)) E . determines a unique locally o-convex topology, 2F ; moreover 2F D( ~)(resp.
44 (c) Let IVn : n ~ 1 1 consisting of circled all n ) I . fz 9 V Let f G2 By the assertion (a), f f 9 V ~-I ~ K ~-i G such that (n) and let E G such that f G~ ~here @ It is clear that completeness of n # @~ because G has been chosen, select is the vector subspace of I~=If k : n ~ I~ X n+• C V for n E is not finite generated by [-f~, fl] U [-fz, fz]o Gz ~ G . After f 9 V G (~ K such that n+l generated by U~= [-fk, fk] is a ~-Cauchy sequence in implies that there exists @ is finite dimensiomal, is finite dimermional, so 2) + V n+i for denote the vector subspaee of be the vector subspace of % 0 is not finite dimensional.
K , k I~(~) * ~ ( ~ ) It is not hard to show that Pk is decomposable if and only E ~ : x,~ on and g and if K = G ~ C . Then for all b y setting ~ ~, ~ - ~ is a seminarm on ~(u) ~ ~(u) C is a positive sublinear functional defined then we can define a functional ~(~) : i ~ E . 11) m~ch that u r K . 6) K = @/~ C o__n G Lemma. and le~ k Le t @ b e a d e c o m p o s a b l e s u b s p a c e 0s be a monotone sublinear fu~tional on K . (E, C) , l e t Define by settin~ ~(x) = i ~ I~(x~) . ~ ( % ) : ~,x ~ ~,x~-% =x~ 9 (x~ ~).
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