I am looking for some good references that introduce functional derivatives in a quick but rigorous way. Any suggestions?
In addition, I saw somewhere that functional derivatives are related to Frechét derivatives. Is this accurate? How are they related?
Meanwhile someone could add something about the references, let me tell you how the Frechét technique is used to get the famous Euler-Lagrange condition with a classical example.
Let us suppose the you have a integral functional like $$A=\int_I{\cal L}(x,y,y')dx$$ where $y=y(x)$.
Now to mimic Frechét procedure we take $$A_{\varepsilon}=\int_I{\cal L}(x,y+\varepsilon h,y'+\varepsilon h')dx.$$ Employ expansions to get $${\cal L}(x,y+\varepsilon h,y'+\varepsilon h')={\cal L}(x,y,y')+\varepsilon h\frac{\partial{\cal L}}{\partial y}+\varepsilon h'\frac{\partial{\cal L}}{\partial y'}+R(\varepsilon^2,x). $$
Upon some basic manipulation you will get $$\frac{A_{\varepsilon}-A}{\varepsilon}= \int_I h\left(\frac{\partial{\cal L}}{\partial y}-\frac{d}{dx}\frac{\partial{\cal L}}{\partial y'}\right)dx+\frac{1}{\varepsilon}\int_IR(\varepsilon^2,x)dx,$$ So, under some mild conditions, in the limit we arrive at $$\lim_{\varepsilon\to0}\frac{A_{\varepsilon}-A}{\varepsilon}= \int_I h\left(\frac{\partial{\cal L}}{\partial y}-\frac{d}{dx}\frac{\partial{\cal L}}{\partial y'}\right)dx,$$ for all function $h=h(x)$, which if they obey $h|_{\partial I}=0$ then, asking for $\lim_{\varepsilon\to0}\frac{A_{\varepsilon}-A}{\varepsilon}=0$ in order to be a detector of extreme data $(x,y,y')$, one is led to the fact that $$\int_I h\left(\frac{\partial{\cal L}}{\partial y}-\frac{d}{dx}\frac{\partial{\cal L}}{\partial y'}\right)dx=0,$$ for all suitable $h$, then is needed that $$\frac{\partial{\cal L}}{\partial y}-\frac{d}{dx}\frac{\partial{\cal L}}{\partial y'}=0,$$ to manipulate and find $y$ the extremizing function.