While sightseeing aspects of Calculus of Variations, the following fact elludes me: there is a plethora of new definitions which seem redundant to me. This phenomenom happens, of course, with other subjects: for instance, one can argue that a vector space is a module over a field instead of making a "new" definition for a vector space (this is not so good of an example due to one being often introduced to vector spaces before modules, but it gets my core idea). However, when such phenomena happens in these other cases, there usually is a nice reference in the literature which makes the correspondence of definitions clear. But in all references on calculus of variations I've seen, a "variation" is a new object that is defined, and I can't see why one should not regard this as simply a case of Fréchet-derivation.
This happens even if one take a path-space of paths connecting a point $a$ to $b$, for instance. Let's consider the space $C^1([0,1], \mathbb{R}^n, a,b)$ the space of $C^1$ paths with initial point $a$ and endpoint $b$. This is an affine space over the normed vector space $C^1([0,1], \mathbb{R}^n, 0,0)$, so we have a bona-fide Fréchet-derivative, and hence we can talk about critical points. The "variations" are simply elements of the vector space.
Therefore, my questions are: Am I missing something? More precisely, is my point of view lacking or incorrect in some aspect?
If not, why isn't this approached in this way?
Specialists in the calculus of variations don't necessarily consider it a subfield of functional analysis. They are entitled to using terminology of their own choice. In a similar spirit, Halmos objected to the logicians' use of terms like interpretation (of a theory in a model), and sought to replace it by homomorphism in a suitably defined type of algebraic formalism. Ultimately his work in polyadic algebras proved to be of little consequence.
Replacing variations by Frechet derivative may be useful if one can then proceed to apply general results about Frechet derivatives and get meaningful consequences for variational calculus; moreover it is quite possible that such results do exist. However, replacing a finite framework by an infinite one usually requires justification. If you take a look at my postings you will notice that I am not opposed to infinity :-) but the question of motivation has to be addressed.