I always thought that real analysis deals with analysis on the real line, eventually on the Euclidean space $\mathbb R^n$.
But why does someone have to label a course as real analysis when it is obviously an generalization and extension of the above to functions on more abstract spaces? Isn't it functional analysis then?
Take $L^p$ space, for example. How can it be in both real and functional analysis?
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$\begingroup$Let's take $L^p$ space for example: how can it be in both real and functional analysis?
If one uses "$L^p$" as notation for the set of functions that are Lebesgue integrable to the $p$th power, and proceeds to study the properties of individual functions in this set, this is Real Analysis. The mode of thinking here is not much different from studying the set $R[a,b]$ of Riemann-integrable functions on $[a,b]$.
If one introduces norm structure on the $L^p$ space, induces topology with it, and proceeds to study continuous linear functionals, uniform convexity, etc, this is Functional Analysis. The focus shifts from the properties of individual functions to the structure of the space as a whole.
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