Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Using boundaries to find smooth norms   The aim of this paper is to present a tool used to show that certain Banach
spaces can be endowed with $C^k$ smooth equivalent norms. The hypothesis uses
particular countable decompositions of certain subsets of $B_{X^*}$, namely
boundaries. Of interest is that the main result unifies two quite well known
results. In the final section, some new corollaries are given.

5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat
28
The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
29
The categories of this paper are: funct-an
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
16
The categories of this paper are: stat