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Bounded sets structure of C p X and quasi‐ ( D F ) ‐spaces
Author(s) -
Ferrando Juan Carlos,
Gabriyelyan Saak,
Ka̧kol Jerzy
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800085
Subject(s) - mathematics , bounded function , space (punctuation) , linear subspace , metrization theorem , countable set , pure mathematics , tychonoff space , interpolation space , locally convex topological vector space , discrete mathematics , compact open topology , hausdorff space , mathematical analysis , topological space , separable space , functional analysis , philosophy , linguistics , biochemistry , chemistry , gene
Abstract For wide classes of locally convex spaces, in particular, for the spaceC p ( X )of continuous real‐valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of ( D F ) ‐spaces have led us to introduce quasi‐ ( D F ) ‐spaces, a class of locally convex spaces containing ( D F ) ‐spaces that preserves subspaces, countable direct sums and countable products. Regular ( L M ) ‐spaces as well as their strong duals are quasi‐ ( D F ) ‐spaces. Hence the space of distributionsD ′ ( Ω )provides a concrete example of a quasi‐ ( D F ) ‐space not being a ( D F ) ‐space. We show thatC p ( X )has a fundamental bounded resolution if and only ifC p ( X )is a quasi‐ ( D F ) ‐space if and only if the strong dual ofC p ( X )is a quasi‐ ( D F ) ‐space if and only if X is countable. If X is metrizable, thenC k ( X )is a quasi‐ ( D F ) ‐space if and only if X is a σ‐compact Polish space.