Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A note on closedness of the sum of two closed subspaces in a Banach
  space   Let $X$ be a Banach space, and $M,N$ be two closed subspaces of $X$. We
present several necessary and sufficient conditions for the closedness of $M+N$
($M+N$ is not necessarily direct sum).

6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin
18
The categories of this paper are: econ
28
The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
21
The categories of this paper are: adap-org