By Yau-Chuen Wong (auth.)
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Extra info for The Topology of Uniform Convergence on Order-Bounded Sets
Example text
1). 2), we get the required equality. In particular, we have, for any (ut~ (A)) ~ C (A), that Ki'-~ %(u,(A)) Let by a family Then 6• (E, C, 9) P =q(Z~ (A)U) . 1) and a well-known result, with hhe relative topology induced by ~ Denoting by 9=D (&~
E, ~) i_~n E . Let (E, C, ~) the completion of Then ~ , ~, ~) be a metrizable locally-solid (E, 2) , and by ~ the is a complete metrizable locally solid space. We are now going to cor~ider the duality of local o-convexity and local decomposability. space. y strict be an ordered topological vector ~-cone in (E, 2) if for ar~ 36 -bounded set B exists a subset A in E of and any E which is absorbed by BCA~C It is clear that a ~ S-cone in C (E, is a strict 9 -oo~ in ~-cone in ~) exists A (E, II " II) ~ ) there such that is a locally strict (E, C, II " II) , if and only if it is a locally strict (E, Then k > 0 Let (E, C, .
The such that f =Z" f . k=~ k Clearly [-fk' f ~ C [--f, f] for all k ~ I . contained in a finite dimensional subspaee of assertion (a) . Let on E Therefore (E, C) E ~ G [-f, f] is not which contradicts the must be finite dimensional. be a Riesz space and let such that the lattice operations are seen that Therefore ~ be a locally convex topology ~-oontinuous. is a locally decomposable topology. 14) of Chap. 3]. In the final chapter we shall study this topology detail, ar~ sb~]] give other characterizatiorm some speci~l type mappings.