Thanks so much for your answer! It helped me a lot! Just to clarify: you new variables $:T^{k}(x):_{V}$ are, again, formal and do not have mathematical meaning right? But the formulas we get from it have meaning?

Do you know any good reference on this subject?

Thanks for the answer. The problem is that I can not find any other reference on this subject. Physicists use requirement (1) as a definition of functional derivative and they are not rigorous só that there is no good physics text where this is discussed in detail. On the other hand, I know a bit about Fréchet and Gâteaux derivatives, which are well-defined and etc but I don't know how to connected with the physicists vision of functional derivative.