Advanced Analysis Study Guide: Measure & Functional

If you wish to follow this guide, you should be familiar with analysis on $\mathbb{R}$ and $\mathbb{R}^n$. See my previous insight for the list of prerequisite topics and book suggestions: https://www.physicsforums.com/insights/self-study-analysis-part-intro-analysis/

You should also be comfortable with linear algebra; see my insight on that: https://www.physicsforums.com/insights/self-study-algebra-linear-algebra/

We will now take one step further. While analysis on $\mathbb{R}$ and $\mathbb{R}^n$ is exciting, it is only the tip of the iceberg. If you learned analysis on $\mathbb{R}$ and $\mathbb{R}^n$ well, you should have no difficulty at all. Most of the material you will encounter are careful but powerful generalizations of the classical context.

Central at this stage of your studies are the concepts of metric spaces, normed spaces, and Hilbert spaces on the one hand, and measure theory on the other. These form the backbone of almost all other types of analysis. The best place to start, in my view, is with

Carothers – Real Analysis

http://www.amazon.com/Real-Analysis-N-L-Carothers/dp/0521497566

This is a gem of a book. It motivates new concepts carefully and provides substantial intuition, motivation, and guidance at every step. The exercises are excellent: many problems are given and some are tagged as “must do” while others are optional.

This book treats:

• Metric spaces
• Function spaces
• Measure theory on $\mathbb{R}$

The coverage is thorough and deep. The only minor limitation is that measure theory on more general spaces is not developed here, but the book is already substantial as it stands.

In the metric spaces section, you will learn:

• Open and closed sets
• Continuity
• Completeness
• Compactness
• Connectedness
• Category theorems

Another book that covers metric spaces very thoroughly is O’Searcoid, Metric Spaces: http://www.amazon.com/Metric-Spaces-Springer-Undergraduate-Mathematics/dp/1846283698/

In the function spaces section, you will encounter:

• Uniform convergence
• Fourier series
• Stone–Weierstrass theorem
• Ascoli–Arzelà theorem
• Bounded variation
• Stieltjes integration

And finally, in the measure theory section, you will study:

• Lebesgue measure
• Measurable functions
• Lebesgue integral
• Differentiation theory

Once you have finished Carothers, it is time to study measure theory in more depth. For that I recommend:

Jones – Lebesgue Integration on Euclidean Space

http://www.amazon.com/Lebesgue-Integration-Euclidean-Bartlett-Mathematics/dp/0763717088

Jones gives a deep but intuitive development of measure theory in $\mathbb{R}^n$ and sometimes addresses general measure spaces. Where many texts rush the construction of Lebesgue measure, Jones proceeds carefully and cleanly. The book also discusses several applications of measure theory within analysis.

This book covers:

• Lebesgue measure on $\mathbb{R}^n$
• Invariance of Lebesgue measure and a formal proof of why the determinant measures volume
• Borel sets and measurable functions
• Lebesgue integration over $\mathbb{R}^n$
• Fubini–Tonelli theorem
• The Gamma function
• $L^p$ spaces
• Convolutions
• Fourier theory on $\mathbb{R}^n$
• Differentiation theory on $\mathbb{R}^n$ and $\mathbb{R}$

It is useful to complement Jones with a text that treats these topics on general measure spaces. For that I highly recommend:

Bartle – The Elements of Integration and Lebesgue Measure
http://www.amazon.com/Elements-Integration-Lebesgue-Measure/dp/0471042226

Bartle covers much of the same territory as Jones but from the more abstract perspective. It is a good idea to work through Bartle and Jones concurrently.

The book addresses:

• Measurable functions
• Measures
• Integral
• Integrable functions
• $L^p$ spaces
• Different kinds of convergence
• Decomposition of measures
• Generation of measures
• Product measures
• Outer measure
• Measurable sets
• Approximation of measurable sets
• Additivity
• Non-Borel sets and non-measurable sets

After Bartle you will have sufficient familiarity with measure theory to read most advanced analysis texts. I should comment that the Lebesgue integral is not the most general integral: on $\mathbb{R}$ there is the Henstock–Kurzweil (gauge) integral, which is stronger in some respects. Studying it is optional, but if you wish to explore it I recommend Bartle, A Modern Theory of Integration: http://www.amazon.com/Modern-Integration-Graduate-Studies-Mathematics/dp/0821808451

At this stage it is also a good idea to begin functional analysis. A standard, elementary introduction is

Kreyszig – Introductory Functional Analysis with Applications
http://www.amazon.com/Introductory-Functional-Analysis-Applications-Kreyszig/dp/0471504599

This book does not require measure theory, so you can read it after the metric and function spaces parts of Carothers (though it is also natural to study Kreyszig alongside Carothers). If you have done measure theory already, Kreyszig remains valuable; it is a useful exercise to try to extend the theorems there to more general contexts.

Kreyszig covers:

• Metric spaces
• Normed spaces and Banach spaces
• Inner product spaces and Hilbert spaces
• Fundamental theorems for normed and Banach spaces
• Banach fixed-point theorem and applications to linear, differential, and integral equations
• Approximation theory
• Spectral theory of linear operators in normed spaces
• Compact linear operators on normed spaces and their spectrum
• Spectral theory of bounded self-adjoint linear operators
• Unbounded linear operators in Hilbert space
• Unbounded linear operators in Quantum Mechanics

As you see, many topics overlap with Carothers, but they are treated here from a different perspective. The book ends with a section on quantum mechanics, which helps to motivate functional analysis. Aside from analysis, be comfortable with linear algebra: functional analysis generalizes linear algebra to infinite-dimensional settings.

The next recommended steps are complex analysis and topology, which I will detail in another insight.