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?

Assume $f\colon\mathbb{R}^n\to\mathbb{R}^m$ has continuous partial derivatives of the first order. Then it is a standard result that $f$ is Fréchet differentiable, i.e., $$f(x+h)=f(x)+f'(x)h+o(\lvert h\rvert)$$ where $f'(x)$ is a linear map $\mathbb{R}^n\to\mathbb{R}^m$. Moreover, the matrix of this map is what you would expect, consisting of the partial derivatives of the component functions.
