Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Test-space characterizations of some classes of Banach spaces   Let $\mathcal{P}$ be a class of Banach spaces and let
$T=\{T_\alpha\}_{\alpha\in A}$ be a set of metric spaces. We say that $T$ is a
set of {\it test-spaces} for $\mathcal{P}$ if the following two conditions are
equivalent: (1) $X\notin\mathcal{P}$; (2) The spaces $\{T_\alpha\}_{\alpha\in
A}$ admit uniformly bilipschitz embeddings into $X$.
  The first part of the paper is devoted to a simplification of the proof of
the following test-space characterization obtained in M.I. Ostrovskii
[Different forms of metric characterizations of classes of Banach spaces,
Houston J. Math., to appear]:
  For each sequence $\{X_m\}_{m=1}^\infty$ of finite-dimensional Banach spaces
there is a sequence $\{H_n\}_{n=1}^\infty$ of finite connected unweighted
graphs with maximum degree 3 such that the following conditions on a Banach
space $Y$ are equivalent:
  (A) $Y$ admits uniformly isomorphic embeddings of $\{X_m\}_{m=1}^\infty$;
  (B) $Y$ admits uniformly bilipschitz embeddings of $\{H_n\}_{n=1}^\infty$.
  The second part of the paper is devoted to the case when
$\{X_m\}_{m=1}^\infty$ is an increasing sequence of spaces. It is shown that in
this case the class of spaces given by (A) can be characterized using one
test-space, which can be chosen to be an infinite graph with maximum degree 3.

16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph