There is something I do not understand well about the definition of the functional derivative. In the wikipedia page http://en.wikipedia.org/wiki/Functional_derivative it says:
1) This definition readily accepts that the functional derivative can be written using the inner product for $L^2$ Hilbert space. But according to the Riesz representation theorem, this should be valid only for a bounded linear functional on the $L^2$ Hilbert space. So as long as $[\dfrac{d}{d\epsilon} F[\rho + \epsilon \phi]]_{\epsilon=0}$ is a bounded linear functional for $\phi \in L^2$, then this definition is true. Is this correct?
2)The inner product definition is used with the dirac delta function in the same Wikipedia page like that:
But according to another resources the evaluation functional $F[\rho] = \rho (r)$ is linear but not bounded in $L^2$ and such the Riesz representation theory should not work in this case. But we can derive $[\dfrac{d}{d\epsilon} F[\rho + \epsilon \phi]]_{\epsilon=0}=\int \delta (r-r') \phi(r')dr'$ where we have obtained an inner product form from which we can read $\delta(r-r')$ as the functional derivative. So the representation theorem seems to "work". How is this justified, am I missing something?