When starting something as challenging as learning graduate-level mathematics, it’s useful to have an idea of the expected study time for mathematical ideas. It’s very easy to deceive ourselves as to how much time is required and, more particularly, how little time we have actually spent studying.
This is particularly the case when self-studying, as we are not governed by the expected pace of a tutor and a given syllabus. It’s easy to get impatient to move on, or frustrated at difficult exercises. One way to “calibrate” materials such as books into familiar levels is to use course outlines from universities. For example, the Course Materials Hub of the Oxford mathematics degree programme provides a summary and often notes for all its courses.
For example, to learn Functional Analysis, there are the following courses over four years, from Prelims (first-year undergraduate) to Part C (Master’s level):
- M2 Analysis I, II, III
- A2.1 Metric Spaces + A2.2 Complex Analysis + A4 Integration
- B4.1 + B4.2 Functional Analysis 1 + 2
- C4.1 Further Functional Analysis
So a book such as Functional Analysis by Joseph Muscat roughly corresponds to six courses: A2.1 (part I of the book), A2.2 (Chapter 12), A4 (Chapter 9.2), B4.1 (most of Part I), B4.2 (remainder of Part I) and C4.1 (parts of Part III), with some differences in the advanced material.
Assuming some kind of background in analysis, this could mean at least 5 or 6 courses worth of material up to Master’s level: at 150 hours per course, that’s 750-900 hours of study for a typical student. 900 hours for one book may seem a lot - however, with 434 pages, that’s around 2 hours a page, including thinking through 1000 examples and exercises.
Maths should be absorbed slowly.