Calculus stay to Real Analysis as $x$ stay to Functional Analysis Hi guys i had a look to book which treat the subject of Calculus (of course...) Analysis and Functional Analysis.
Is that correct to state that Calculus is more focused on "computing" while "Analysis" is more focused on theorem proving?
If yes what is the "calculus" version of Functional Analysis?
I had a glance to some book of functional analysis and most of them are pretty abstract and much focused on theorem proving, i mean... while in calculus the theory is theorem too the exercises are most focused on computation. Is there something similar for functional analysis?
I don't know all the theory of functional analysis, Rudin as instance speaks a bit of spectral theory, distributions, bounded and unbounded operator is there a "calculus" branch for this stuff?
 A: In my opinion, the "calculus side" of functional analysis is... linear algebra. Functional analysis deals with vector spaces endowed with a topology (normed spaces, topological vector spaces), and the most elementary example of such spaces are finite-dimensional vector spaces. 
Linear operators on finite-dimensional spaces are represented by matrices, spectral theory is the elementary theory of diagonalization of various kinds of matrices.
A: In the same way that calculus was invented for computations in basic physics (kinematics, etc), and analysis was a rigorization of the heuristics, I think much of functional analysis was a rigorization of fancier computations in more modern physics, especially quantum theory.
Even before that, around 1900 rigorous treatment (such as it was) of ODEs and PDEs went by way of integral equations, which were easier to be confident of, as bounded operators.
In the 1920s, the basic models of operators $P,Q$ on a Hilbert space, with $PQ-QP=1$, could not possibly be correct with bounded operators. This did not necessarily deter physicists, but was a bit unnerving for mathematicians. Stone's and von Neumann's ~1930 rigorous treatment of unbounded operators on Hilbert spaces was visibly motivated by this.
Similarly, Heaviside's, and even more so Dirac's, use of "generalized functions" certainly was a big motivation in L. Schwartz' conception of "distributions", their Fourier transforms, and so on, to rigorize the wonderful heuristics that were so fruitful in physics.
(Similarly, many aspects of PDE are obviously motivated by physical problems: Navier-Stokes equation...)
