Fréchet derivative I have been doing some self study in multivariable calculus.
I get the geometric idea behind looking at a derivative as a linear function, but analytically how does one prove this?
I mean if $f'(c)$ is the derivative of a function between two Euclidean spaces at a point $c$ in the domain... then is it possible to explicitly prove that $$f'(c)[ah_1+ bh_2] = af'(c)[h_1] + bf'(c)[h_2]\ ?$$ I tried but I guess I am missing something simple.
Also how does the expression
$f(c+h)-f(c)=f'(c)h + o(\|h\|)$
translate into saying that $f'(c)$ is linear?

It is not true that $f'(c_1+c_2)=f'(c_1)+f'(c_2)$ in general.
However, the derivative $f'(c)$ is the matrix of the differential $df_c$ and the expression which you write shows why $df_c(h) = f'(c)h$ is linear in the $h$-variable. It is simply matrix multiplication and all matrix multiplication maps are linear. The subtle question is if $f'(c)$ exists for a given $f$ and $c$.
Recall that $f: \mathbb{R} \rightarrow \mathbb{R}$ has a derivative $f'(a)$ at $x=a$ if
$$ f'(a) = \lim_{ h \rightarrow 0} \frac{f(a+h)-f(a)}{h} $$
Alternatively, we can express the condition above as
$$ \lim_{ h \rightarrow 0} \frac{f(a+h)-f(a)-f'(a)h}{h} =0.$$
This gives an implicit definition for $f'(a)$. To generalize this to higher dimensions we have to replace $h$ with its length $||h||$ since there is no way to divide by a vector in general. Recall that  $v \in \mathbb{R}^n$ has $||v|| = \sqrt{v \cdot v}$. Consider $F: U \subseteq \mathbb{R}^m \rightarrow \mathbb{R}^n$ if $dF_{a}: \mathbb{R}^m \rightarrow \mathbb{R}^n$ is a linear transformation such that 
$$ \lim_{ h \rightarrow 0} \frac{F(a+h)-F(a)-dF_a(h)}{||h||} =0 $$
then we say that $F$ is differentiable at $a$ with differential $dF_a$. The matrix of the linear transformation $dF_{a}: \mathbb{R}^m \rightarrow \mathbb{R}^n$ is called the Jacobian matrix $F'(a) \in \mathbb{R}^{m \times n}$ or simply the derivative of $F$ at $a$.  It follows that the components of the Jacobian matrix are partial derivatives of the component functions of $F = (F_1,F_2, \dots , F_n)$
$$ J_F = \left[ \begin{array}{cccc}
           \partial_1 F_1  & \partial_2 F_1 & \cdots & \partial_n F_1 \\
           \partial_1 F_2  & \partial_2 F_2 & \cdots & \partial_n F_2 \\
               \vdots      & \vdots         & \vdots  & \vdots \\
           \partial_1 F_m  & \partial_2 F_m & \cdots & \partial_n F_m \\
           \end{array} \right] = \bigl[\partial_1F \ | \ \partial_2F \ | \cdots | \ \partial_nF\ \bigr] =
\left[ \begin{array}{c}
           (\nabla F_1)^T \\ \hline
           (\nabla F_2)^T \\ \hline
               \vdots  \\ \hline
           (\nabla F_m)^T
           \end{array} \right].
$$
