Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Compactness in normed spaces: a unified approach through semi-norms   In this paper we prove two new abstract compactness criteria in normed
spaces. To this end we first introduce the notion of an equinormed set using a
suitable family of semi-norms on the given normed space satisfying some natural
conditions. Those conditions, roughly speaking, state that the norm can be
approximated (on the equinormed sets even uniformly) by the elements of this
family. As we are given some freedom of choice of the underlying semi-normed
structure that is used to define equinormed sets, our approach opens a new
perspective for building compactness criteria in specific normed spaces. As an
example we show that natural selections of families of semi-norms in spaces
$C(X,\mathbb{R})$ and $l^p$ for $p\in[1,+\infty)$ lead to the well-known
compactness criteria (including the Arzel\`a-Ascoli theorem). In the second
part of the paper, applying the abstract theorems, we construct a simple
compactness criterion in the space of functions of bounded Schramm variation
and provide a full characterization of linear integral operators acting from
the space of functions of bounded Jordan variation to the space of functions of
bounded Schramm variation in therms of their generating kernels.

21
The categories of this paper are: adap-org
8
The categories of this paper are: hep-ex
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th