Was led to the paths of real analysis and functional analysis both are difficult and fascinating. This blog is to figure out how to tread through. According to Bruce Balden in Quora,

Real Analysis generally comes first, although there’s no shortage of topics which can fall in both a Real Analysis and a Functional Analysis.

The standard reference works by the same author are the following books, all by Walter Rudin

Principles of Mathematical Analysis (aka “baby Rudin”)
Real and Complex Analysis
Functional Analysis

The first book is widely taught at the 3rd year undergraduate level. The third book is widely taught at the 1st year graduate level. The second book is less popular, but has a comparable place to the mainstay which is H.L. Royden’s Real Analysis. Since I have all 4 books at hand, let’s peek at the tables of contents:

Baby Rudin

• Real and Complex number systems
• Basic Topology
• Numerical Sequences and Series
• Continuity
• Differentiation
• Rieeman-Stjieljes Integral
• Sequences and Series of Functions
• Some Special Functions (especially the gamma function)
• Functions of Several Variables
• Integration of Differential Forms (generalized Stokes theorem)
• Lebesgue Theory

Note that the last two chapters are frequently not taught. They weren’t included in the class I took
Now for Royden:

• Set Theory
• Theory of Functions of a Real Variable
• Lebesgue Measure
• The Lebesgue Integral
• Differentiation and Integration
• The Classical Banach Spaces (a topic that recurs later in functional analysis)
• Metric spaces (covered in less generality back in Baby Rudin)
• Topological Spaces (again in greater detail than back in Baby Rudin)
• Compact Spaces — a definite topic of overlap with functional analysis, but the most interesting results here are marked as optional — skipped the first time through
• Banach Spaces — some advanced functional analysis subtopics, but all marked as optional/advanced
• Measure and Integration — general approach to these topics instead of Lebesgue’s treatment for the real line. Includes the crucial LpLp and the Radon-Nikodym theorem, both topics from functional analysis[/math]
• more stuff esp the Daniell Integral which is a functional approach to the Lesbesgue integral.
• A crucial and elementary functional analysis result never mentioned in Royden: the Hahn-Banach theorem, the stepping point to move the operations to spaces of infinite dimension.

Middle Rudin (Real and Complex Analysis)

• Abstract Integration
• Positive Borel Measures — used over in probability theory
• L^p spaces
• Elementary Hilbert Space Theory — much more in Senior Rudin
• Examples of Banach Space Techniques
• Complex Measures — a topic ignored in many other books. Rudin likes to unify real and complex analysis but not everyone is so eager to do so.
• Integration on Product Spaces — a clearer explanation of this crucial topic than found anywhere else I can think of.
• Differentiation
• Fourier Transform
• Elementary Theory of Homophoric Functions — stripped of the strange language this is complex variable theory! Are we in the right course?? Rudin unifies them in a way other authors have not.
• Harmonic Functions — reason Rudin is unifying the real and complex worlds is that they collide in Harmonic Function theory. The real and imaginary parts of a holomorpic function are each harmonic functions of 2 real variables.
• Much more in this book, some of the theory of PDE’s and some from complex analysis.

Finally, senior Rudin’s topic list is similar but quite different

• Topological Vector Spaces
• Completeness — includes the open mapping theorem
• Convexity — includes the Hahn-Banach theorem
• Duality in Banach Spaces — this is where true functionals, linear objects which act on target to produce a real number, enter the picture.
• Test Functions and Distributions — various varieties of distributions are of great utility in the theory of partial differential equations.
• Banach Algebras and Spectral Theory — this corner of functional analysis helps to solve the Heisenberg version of the Schrodinger equation in quantum physics.