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Let $\Lambda$ be an open set in Banach space $E$, $M(x)$ for $x\in \Lambda$ asubspace in $E$, and $\mathcal F=\{M(x)\}_{x\in\Lambda}$. In this paper, weintroduce 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 whereespecially, $\mathrm{dim}\, M(x)$ maybe $\infty$. Using the Frobenius theorem,we prove a theorem in advanced calulus that presents several $c^1$ integrablefamily $\mathcal F$ at a point with $\mathrm{dim}\, M(x) =\infty$ and trivialco-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 followingtheorem is proved using Frobenius theorem: if $A\in\Lambda$ is doublesplitting, then $\mathcal F$ at $A$ is smooth integrable where $J(A,E_*)$ isnon-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$\Phi$. Due to the need of advanced science and technologies, such as advancednano-technology, the best of information transit, they urgently need theextremum principle under non-regular constraints, we give such an extremumprinciple.  Consider the $c^1$ nonlinear functional $f$ from $U$ to $(-\infty , \infty)$. Let $S$ be $c^1$ submanifold in $U$. we prove the following extremumprinciple of $f$ under the constraint $S$: if $x_0$ is an extreme point of $f$under the constraint $S$, then $ T_{x_0}S\subset N(f'(x_0)).$ The principleestablishes a broad mathematical basis of optimization theory under thenon-regular and regular constraints.

Frobenius Theorem in Banach Space and an Extremum Principle of Real Functional under the Constraints of $c^1$ Submanifold in Banach Space