Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fundamental bounded resolutions and quasi-$(DF)$-spaces   We introduce a new class of locally convex spaces $E$, under the name
quasi-$(DF)$-spaces, containing strictly the class of $(DF)$-spaces. A locally
convex space $E$ is called a quasi-$(DF)$-space if (i) $E$ admits a fundamental
bounded resolution, i.e. an $\mathbb{N}^{\mathbb{N}}$-increasing family of
bounded sets in $E$ which swallows all bounded set in $E$, and (ii) $E$ belongs
to the class $\mathfrak{G}$ (in sense of Cascales--Orihuela). The class of
quasi-$(DF)$-spaces is closed under taking subspaces, countable direct sums and
countable products. Every regular $(LM)$-space (particularly, every metrizable
locally convex space) and its strong dual are quasi-$(DF)$-spaces. We prove
that $C_{p}(X)$ has a fundamental bounded resolution iff $C_{p}(X)$ is a
quasi-$(DF)$-space iff the strong dual of $C_{p}(X)$ is a quasi-$(DF)$-space
iff $X$ is countable. If $X$ is a metrizable space, then $C_k(X)$ is a
quasi-$(DF)$-space iff $X$ is a Polish $\sigma$-compact space. We provide
numerous concrete examples which in particular clarify differences between
$(DF)$-spaces and quasi-$(DF)$-spaces.

31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
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The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
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The categories of this paper are: econ
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The categories of this paper are: hep-lat
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an