Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Banach limits -- Some new thoughts and perspectives   The existence of a Banach limit as a translation invariant positive
continuous linear functional on the space of bounded scalar sequences which is
equal to 1 at the constant sequence (1,1,...,1,...) is proved in a first course
on functional analysis as a consequence of the Hahn Banach extension theorem.
Whereas its use as an important tool in classical summability theory together
with its application in the existence of certain invariant measures on compact
(metric) spaces is well known, a renewed interest in the theory of Banach
limits has led to certain applications which have opened new vistas in the
structure of Banach spaces. The paper is devoted to a discussion of certain
developments, both classical and recent, surrounding the theory of Banach
limits including the structure of the set of Banach limits with special
emphasis on certain aspects of their applications to the existence of certain
invariant measures, vector valued analogues of Banach limits, functional
equations and in the structure theory of Banach spaces involving the existence
of selectors of certain multi-valued mappings into the metric space of
non-empty, convex, closed and bounded subsets of a Banach space with respect to
the Hausdorff metric. The paper shall conclude with a brief description of some
recent results of the author on the study of simultaneous continuous linear
operators (linear selections) involving Hahn Banach extensions on spaces of
Lipschitz functions on (subspaces of) Banach spaces. Some open problems that
naturally arise in the study have also been included.

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The categories of this paper are: physics
13
The categories of this paper are: quant-ph
23
The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
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The categories of this paper are: cs
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The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
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The categories of this paper are: math-ph
35
The categories of this paper are: mtrl-th