Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Frobenius Theorem in Banach Space   Let $\Lambda$ be an open set in Banach space $E$, $M(x)$ for $x\in \Lambda$ a
subspace in $E$, and $\mathcal F=\{M(x)\}_{x\in\Lambda}$. In this paper, we
introduce the concept of the co-final set $J(x_0,E_*)$ for $\mathcal F$ at
$x_0\in \Lambda$, then prove Frobenius theorem in Banach space where
especially, $\mathrm{dim}\, M(x)$ maybe $\infty$. Using the Frobenius theorem,
we prove a theorem in advanced calulus that presents several $c^1$ integrable
family $\mathcal F$ at a point with $\mathrm{dim}\, M(x) =\infty$ and trivial
co-final set. Let $\Lambda=B(E,F)\setminus\{0\},\ M(X)=\{T\in B(E,F) :
TN(X)\subset R(X)\}$, and $\mathcal F=\{M(X)\}_{X\in\Lambda}$. The following
theorem is proved using Frobenius theorem: if $A\in\Lambda$ is double
splitting, then $\mathcal F$ at $A$ is smooth integrable where $J(A,E_*)$ is
non-trivial in general. Specially we have the global result as follows, let
$\Phi$ be any one of $F_k(k<\infty)$ and $\Phi_{m,n}(\min\{m,n\}<\infty)$, then
$\Phi$ is smooth submanifold in $B(E,F)$ and tangent to $M(X)$ at any $X$ in
$\Phi$.

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The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: math-ph
10
The categories of this paper are: q-bio
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The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
9
The categories of this paper are: nlin
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The categories of this paper are: plasm-ph