An engineer would like to read further in details the above mentioned topics, I understand those are two separate topics on their own. But I was looking for a book treating both subjects with good transition. Appreciated.
In my work, I often model the physics of processes. I use the variational (weak relations) method to derive the plurality of numerical solutions to systems of nonlinear and coupled PDEs. During the courses I had taken, there was no rigorous treatment of the underlying mathematics. I would like to grasp how to arrive at generalizations, analysis of functional spaces, linear operators, particularly relations like green identities, or operators and tensor calculus.
That is why I asked for some book that starts with real then ends up with functional analysis with demonstration of the foundations and a presentation of the motivations throughout.