Is it possible to learn functional analysis without real analysis?

Question

Are there parts in functional analysis that would be easier to learn without real analysis or is this just not possible?

When I say functional analysis I mean

1. Lebesgue measure and integral
2. spaces of Lebesgue integrable functions
3. Banach spaces
4. duality
5. bounded linear operator and linear operators in general
6. Hilbert space
7. reproducing kernel Hilbert space
8. non-linear analysis in Banach spaces

Answer 1

In my experience, what differentiates "Analysis" from other math regarding functions and the real or complex numbers etc, is the level of depth and rigor. You could probably learn something in each of the topics you listed, but it would be hard to understand the proofs or justification behind a lot of results. Consequently it's difficult to have a deep or truly meaningful understanding of functional analysis without real analysis experience.

Although there would maybe be more specific issues, like not being familiar with specific definitions and concepts stated in Theorems or otherwise. Again, you could learn these definitions but this boils down into what I said above.

Answer 2

Items 1 and 2 listed by you specifies Lebesgue measure and integrable functions. It will be unwise to try to understand them without having been exposed Riemann integral which is part of standard first course in real analysis. That Lebesgue theory is logically a generalisation and so usual Riemann integration is a special case is correct but only academically so.

Also textbooks of functional analysis are not written for people not exposed to real analysis. So it will be difficult to lern it.

It is possible to do mathematics by communicating with other mathematicians by writing letters (Euler, Newton, Fermat, Gauss did so). But you want a modern internet based forum now. And email and LaTeX all are needed though logically it is not a requirement. Now you decide how you should learn Functional Analysis.