I am confused by the computation of a functional derivative.
Say I have a family of functions parameterized by a (infinite dimensional) vector $\vec{c}$:
$$ S_{\vec{c}}(x)= \sum_n^\infty c^n x^n. \tag{1}$$ Then I can write an "inverse" equation for the coefficients $c^n$ in terms of the following functional:
$$c^n[S]= \frac{1}{n!}\frac{d^n S(x)}{dx^n} \bigg{|}_{x=0}.$$ The map from $\{c^n\}$ to functions of the form in (1) is differentiable with $\frac{\partial S}{\partial c^n}=x^n$ since it is just linear in the coefficients $c^n$. However, when I compute the functional derivative of the inverse I run into trouble. We can write:
$$ \delta c^n[S][\eta] \equiv \int \frac{\delta c^n}{\delta S}(x) \eta(x) dx = \lim_{\epsilon \to 0} \frac{c^n[S+\epsilon \eta]-c^n[S]}{\epsilon} = \frac{1}{n!} \eta^{(n)}(0).$$
Now physicists (of which I am one) like to take the test function $\eta$ to be $\delta(x)$, a delta function at some point x (this might not be well-defined and could be the origin of my confusion). I.e. we define
$$\frac{\delta c^n[S]}{\delta S(x)} \equiv \delta c^n[S][\delta(x)] = \frac{1}{n!} \delta^{(n)}(0).$$
Now depending on how the quantity $$\frac{\delta c^n[S]}{\delta S(x)}$$ appears in an expression this might still be well-defined through the relation
$$\delta^{(n)}[f]=(-1)^{n}\delta[f^{(n)}]$$
However, I am dealing with an expression of the form
$$ \int f(S(x)) \frac{\delta c^m[S]}{\delta S(x)} \frac{\delta c^n[S]}{\delta S(x)} dx$$
which does not appear to be well-defined since it evolves a product of derivatives of delta functions. Is there some better way to handle this expression so that it is meaningful. Is there a different bijection between $\{c^n\}$ and S that is more differentiable?
This expression occurs when I try to perform a change of variable on a metric on the space of functions $S$. For a metric $g_{\mu\nu}$ on a finite dimensional space described by either coorindates $\{c^\mu\}$ or $\{d^i\}$ this would look like
$$ g^{\mu\nu} = \sum_{ij} g^{ij} \frac{\partial c^\mu}{\partial d^i} \frac{\partial c^\nu}{\delta d^j}$$
Here $g_{\mu\nu}$ the metric in $c$ coodinates and $g_{ij}$ the metric in $d$ coordinates. And the metric with upper indices is the metric inverse.