Why do we call Functional Analysis like this? Functional analysis is 'a kind of mathematical analysis' where the object of study are functions. The tool for studying functions are the operators. A specific type of operators are the functionals. My question is why do we call this subject as functional analysis, while the main role here is the operator? And if someone has any information when first is this name used and by who (because I think that the answer may have a historical background)?
 A: Short answer. Because it mainly deals with function spaces. 
As opposed to calculus, where all the norms of Euclidean spaces are equivalent (and all Hausdorff topologies which respect their linear nature), in the case of infinite dimensional linear spaces norms are not equivalent. And Functional analysis is the "analysis" study of infinite (in general) dimensional linear spaces, which are in pretty the most interesting cases for applications, function spaces.  
According to the french wikipedia: L'analyse fonctionnelle est la branche des mathématiques et plus particulièrement de l'analyse qui étudie les espaces de fonctions.
A: The name functional was originally used to refer to functions whose arguments were functions. In this sense an operator on a vector space of functions would be a functional. 
Looking around it appears that the first use of the word functional was in Jacques Hadamard's book "Leçons sur le calcul des variations" which translates to "Lessons in the calculus of variations," which was published in 1910.
