TITLE: Taylor expansion of functional
QUESTION [8 upvotes]: The question asks for the Taylor expansion of a functional. Thus, given a real functional $f(g(x))$, what is the Taylor expansion about a function $h(x)$. What if the function is multi-variate, e.g., $f(x_1,x_2,g(x_1,x_2))$?
I've searched the web for an answer to this, and haven't come up with anything definite. If someone could help, I'd be grateful.

REPLY [10 votes]: This is a physicist answer and is thus not expressed in the better mathematical "vocab", but the Taylor expansion around $x_0$ of a functional $f$ of a function $g(x)$, i.e., of $f[g(x)]$ noted $f[g]$ for simplicity is:
$f[g]=f[g_0]+\int dx \frac{\delta f[g_0]}{\delta g(x)}\Delta g(x)+\frac{1}{2!}\int dx dx^\prime \frac{\delta^2f[g_0]}{\delta g(x^\prime)\delta g(x)}\Delta g(x^\prime)\Delta g(x) + ... $
with $g_0=g(x_0)$ and $\Delta g(x)=g(x)-g(x_0)$