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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:

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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:

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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?

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The definition you mentioned of the functional derivative is valid, whenever $[\frac{\mathbb{d}}{\mathbb{d}\epsilon}F[\rho+\epsilon\phi]]_{\epsilon=0}$ is linear in $\rho$ and can be represented as an integral with some function, which is not necessarily in $L^2$. Hence this linear functional may not be bounded as a functional $L^2 \rightarrow \mathbb{R}$. I remind you that the inner product on $L^2$ is defined only for function in $L^2$ and not for general functions, so in the case where $\frac{\delta F}{\delta \rho}\notin L^2$ you can not use the notion of inner product when referring to the functional derivative.

As for the second part of your post, when we allow the functional derivative to be the delta function, we actually extend this definition since the delta function is not really a function, but rather a linear functional on functions. In this case $[\frac{\mathbb{d}}{\mathbb{d}\epsilon}F[\rho+\epsilon\phi]]_{\epsilon=0}$ is some linear functional that can not be represented as an integration with a function. Hence it is not a bounded linear functional $L^2 \rightarrow \mathbb{R}$, but it is still well-defined using the extended definition.

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