Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Smooth points in operator spaces and some
  Bishop-Phelps-Bollob$\acute{a}$s type theorems in Banach spaces   We introduce the notion of approximate norm attainment set of a bounded
linear operator between Banach spaces and use it to obtain a complete
characterization of smooth points in the space of compact linear operators,
provided the domain space is reflexive and Kadets-Klee. We also apply the
concept to characterize strong BPB property (sBPBp) of a pair of Banach spaces.
We further introduce uniform $ \epsilon- $BPB approximation of a bounded linear
operator and uniform strong BPB property (uniform sBPBp) with respect to a
given family of norm one linear operators and explore some of the relevant
properties to illustrate its connection with earlier studies on
Bishop-Phelps-Bollob$\acute{a}$s type theorems in Banach spaces. It is evident
that our study has deep connections with the study of smooth points in operator
spaces. We obtain a complete characterization of uniform sBPBp for a pair of
Banach spaces, with respect to a given family of norm one bounded linear
operators between them. As the final result of this paper, we prove that if $
\mathbb{X} $ is a reflexive Kadets-Klee Banach space and $ \mathbb{Y} $ is any
Banach space, then the pair $ (\mathbb{X},\mathbb{Y}) $ has sBPBp for compact
operators. Our results extend, complement and improve some of the earlier
results in this context.

35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
16
The categories of this paper are: stat
10
The categories of this paper are: q-bio
20
The categories of this paper are: acc-phys
7
The categories of this paper are: hep-th
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The categories of this paper are: q-fin
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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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: cmp-lg
28
The categories of this paper are: alg-geom