Lagrange multipliers in Banach space I am very familiar with the technique of Lagrange multipliers in optimization problems with constrains in several variables. My question is related to its extension to Banach spaces. Specifically, I would like to  prove the following result which I read on some notes on optimization (without proof)
Theorem: Suppose $f$, $g_1,\ldots,g_m$ are functions in $\mathcal{C}^1(\Omega,\mathbb{R})$, where $\Omega$ is an open set in a (real) Banach space $X$. If $x_0\in\Omega$ is a local minimum for $f$ restricted on $M=\{x\in\Omega: g_1(x)=\ldots=g_m(x)=0\}$, then there exists numbers $\mu$, $\alpha_1,\ldots,\alpha_m$ such that
$$
\mu f'(x_0)+\alpha_1 g_1'(x_0)+\ldots+\alpha_m g_m'(x_0)=0
$$
Moreover, if $\{g'_1(x_0),\ldots,g'_m(x_0)\}\subset X^*$ is a linearly independent, then $\mu\neq0$, and there is a unique $\boldsymbol{\lambda}\in\mathbb{R}^m$ such that
$$
f'(x_0)+\lambda_1 g_1'(x_0)+\ldots+\lambda_m g_m'(x_0)=0.
$$
A reference, or a hint for a proof will be appreciated.
 A: The result in the OP is a first order extension to Banach space of Lagrange multipliers. The following sketch of a proof is based on the surjective theorem which I state at the end.
Sketch of a proof:
Let $U$ be a ball around $x_0$ such that $f(x_0)\geq f(x)$ for all
$x\in U\cap M$. Let $F:U\longrightarrow\mathbb{R}^{n+1}$ be the
function given by
$$
F(x)=(f(x),g_1(x),\ldots,g_n(x))^\top
$$
For any $r>f(x_0)$,  the vector $(r,0,\ldots,0)^\top\notin F(U)$. Hence,
$F(U)$ does not contained any open neighborhood of the point
$(f(x_0),g_1(x_0),\ldots,g_n(x_0))^\top=(f(x_0),0,\ldots,0))^\top$.
Then, $F'(x_0)$ is not surjective. Therefore, the range $V$ of $F'(x_0)$
is a proper subspace of $\mathbb{R}^{n+1}$. Let
$(\mu,\boldsymbol{\lambda})^\top=
(\mu,\lambda_1,\ldots,\lambda_n)^\top$ be nonzero element in
$V^\perp$. Then
$$
\mu f'(x_0)v+\sum^n_{k=1}\lambda_k g'(x_0)v=0
$$
for all $v\in X$,  and the conclusion of the first statement follows.
When $G=\{g'_1(x_0),\ldots,g'_n(x_0)\}$ is linearly
independent, then $\mu\neq0$. Dividing by $\mu$ if necessary, one can
assume $\mu=1$. The uniqueness of $\boldsymbol{\lambda}$ follows
from the linear independence of $G$.

The surjective theorem:
Let $X$, $Y$ be Banach spaces and $\Omega\subset X$ open. Assume that
$F\in\mathcal{C}^1(\Omega,Y)$ and that for some $x_0\in \Omega$, $F'(x_0)$ has a right hand inverse in $\mathcal{L}(Y,X)$. Then, $F(\Omega)$ contains
an open ball around $f(x_0)$.
This is a classic result and can be found in many books on nonlinear analysis. The version I am quoting is from Ward Cheney's Analysis for applied mathematics. David Luenberger's Optimization by vector space methods also has other extensions for Vector spaces with partial orders.
