Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ball generated property of direct sums of Banach spaces   A Banach space $X$ is said to have the ball generated property (BGP) if every
closed, bounded, convex subset of $X$ can be written as an intersection of
finite unions of closed balls. In 2002 S. Basu proved that the BGP is stable
under (infinite) $c_0$- and $\ell^p$-sums for $1<p<\infty$. We will show here
that for any absolute, normalised norm $\|\cdot\|_E$ on $\mathbb{R}^2$
satisfying a certain smoothness condition the direct sum $X\oplus_E Y$ of two
Banach spaces $X$ and $Y$ with respect to $\|\cdot\|_E$ enjoys the BGP whenever
$X$ and $Y$ have the BGP.

20
The categories of this paper are: acc-phys
23
The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org