The Analysis a Beginning Graduate Student Ought to Know

ANALYSISI take this title (with a bit of a modification) from a textbook I used during my first year of my PhD. It was a book by Jonathan Golan entitled The Linear Algebra a Beginning Graduate Student Ought to Know. Now, I did not know most of what was covered in Golan’s book, but I found much of it to be helpful. What this post will discuss is analysis, though. Analysis is this strange creature that tends to catch many grad students off guard. Disclaimer: I am not an analyst. What is here is merely my experience and how I perceive analysis. It is different. It is difficult. It is the foundations of mathematics.

Wolfram Mathworld says,

Analysis is the systematic study of real and complex-valued continuous functions. Important subfields of analysis include calculus, differential equations, and functional analysis. The term is generally reserved for advanced topics which are not encountered in an introductory calculus sequence, although many ideas from those courses, such as derivatives, integrals, and series are studied in more detail. Real analysis and complex analysis are two broad subdivisions of analysis which deal with real-values and complex-valued functions, respectively.

Typically (at least in my experience), in your first year graduate analysis course, you will cover topics like the Real and Complex number systems, basic topology, numerical sequences and series, continuity, differentiation, Riemann-Stieltjes integral, and sequences and series of functions (I took this from the table of contents of Rudin).

Evelyn Lamb just wrote an article On Teaching AnalysisHer post discusses Timothy Gowers’s blog during his time teaching analysis at The University of Cambridge. For learning analysis (and anything in grad school really), it is important to really understand the material completely. To do this, you may need to take a look at more sources. Gowers’s blog is a place to start. You may also want to take a look at the following resources:

If you have already taken analysis, what resources did you find helpful? If you are currently taking analysis, what resources do you wish you have that you cannot find? Happy studying!

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