Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the *spectrum* ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, endowed with some natural topology, say, the topology of uniform convergence on compact sets in $A$. For each $s\in{\tt Spec}[A]$ let us call a *tangent space* in $s$, ${\tt T}_s[A]$, the set of all tangent vectors in $s$, i.e. continuous linear functionals $\tau:A\to{\mathbb C}$ such that
$$
\tau(f\cdot g)=s(f)\cdot\tau(g)+\tau(f)\cdot s(g),\qquad f,g\in A
$$
(again, we endow ${\tt T}_s[A]$ with some natural topology, say, the topology of uniform convergence on compact sets in $A$).

Let $M$ be a Stein manifold and $A={\mathcal O}(M)$, the algebra of holomorphic functions on $M$ (with the usual topology of uniform convergence on compact sets in $M$).

I believe that $$ {\tt Spec}[{\mathcal O}(M)]\cong M, $$ (a homeomorphism of topological spaces), and for each $s\in M$ $$ {\tt T}_s[{\mathcal O}(M)]\cong{\tt T}_s(M) $$ (an isomorphism with the usual tangent space of $M$ in the point $s\in M$), but I can't find a reference. Can anybody help me?

EDIT. Ben McKay gave a reference for the first equality, so it remains to prove the second one. As far as I understand, here everything follows from the following variant of

Hadamard's lemma: Let $s$ be a point on a Stein manifold $M$, and $f_1,...,f_n\in {\mathcal O}(M)$ a sequence of functions which form local coordinates in a neighbourhood of $s$ and $$ f_1(s)=...=f_n(s)=0 $$ (such $f_k$ always exist). Then for every function $f\in{\mathcal O}(M)$ there exist functions $g_1,...,g_m,h_1,...,h_m\in {\mathcal O}(M)$ such that $$ g_1(s)=...=g_m(s)=h_1(s)=...=h_m(s)=0 $$ and $$ f=f(s)+\sum_{k=1}^n\frac{\partial f}{\partial f_k}(s)\cdot f_k+ \sum_{k=1}^m g_k\cdot h_k $$ where $\frac{\partial f}{\partial f_k}(s)$ are partial derivatives in $s$ along the coordinates $f_k$.

That is not true?