Introduction to Real Analysis (Trench)
William F. Trench’s two-term real analysis text for junior and senior mathematics majors. Free edition December 2013, under CC BY-NC-SA 3.0, hosted in Trinity University’s Digital Commons with the LaTeX source (12.8 MB) alongside the PDF.
This closes the spoke’s second named hole (synthesis growth edge 2). Together with hefferon-linear-algebra it takes the corpus past calculus for the first time.
What it covers
Eight chapters:
- The Real Numbers — the completeness axiom and what the calculus sequence assumed.
- Differential calculus of one variable
- Integral calculus of one variable
- Infinite sequences and series
- Real-valued functions of several variables
- Vector-valued functions of several variables
- Integrals of functions of several variables
- Metric spaces
Two supplements are published separately — functions defined by improper integrals, and the method of Lagrange multipliers.
Chapters 5 to 7 cover the same ground as corral-vector-calculus and chapters 10–13 of keisler-elementary-calculus, in the rigorous register. The corpus now holds multivariable calculus three times over from three directions, which is more corroboration than it has of anything else.
Where it stands in the foundational fork
Trench’s own description of the book is that it “fills the gaps left in the development of calculus” — which is the standard account: the calculus course gets the results, analysis supplies the proofs that were skipped. Chapter 1 builds the real numbers from completeness and everything follows from there.
That is exactly the sequence Keisler argues was never revisited after 1960 (synthesis → the foundational fork). Trench does not mention infinitesimals or the alternative; he assumes the ε-δ development the way Corral does.
This is not the source that resolves the fork. The corpus still has no text that argues the standard side deliberately — it now has a second one that assumes it, from a more rigorous author, which is weak evidence for Keisler’s inertia thesis and not evidence against it. The open question stands, and the shape of the answer it needs is sharper: something that considered non-standard analysis and declined it in print.
Why T1
The book was published commercially by Pearson Education before Trench reissued it free, so it carries a publisher’s editorial and review process the corpus’s self-published texts do not. The author is a professor emeritus at trinity-university writing in his own research area. The instructor’s manual stays under all-rights-reserved and is released only to verified faculty, which is a small tell that the free edition is a reissue of a course-adopted book rather than a personal project.
Connections
- author william-trench
- affiliation trinity-university
- fills the analysis hole named in synthesis
- covers keisler-elementary-calculus and corral-vector-calculus ground in the rigorous register
- the ε-δ development infinitesimal-calculus is the alternative to