Axiom of Choice in Mathematical Analysis

Question

I'm an undergraduate student and I'm looking for a book, suitable for self-study, that "explains" the applications of the axiom of choice in mathematical analysis. I'm not familiar with the axiom of choice, so I'd prefer a beginner-level book. If anybody could help me with this, I'd be grateful. I really don't know where to look for something like this.

Answer 1

• Horst Herrlich - The Axiom of Choice

  Definitely worth looking into. It's very light on set theoretic proofs, at least compared to "the competition".

• Eric S. Schechter - The Handbook of Analysis and its Foundations

  Written from a functional analytic point of view. It contains a very thorough investigation of choice-related topic, and of course a lot functional analysis.

• David Fremlin - Measure Theory

  A very dense book, whose last volume is about set theoretic measure theory, and includes a lot of analysis without the axiom of choice, and what we can do and not do without choice.

Regardless to all that, I suggest that you study analysis first, and set theory alongside of it. Just to get used to the language and methods. Then learning about choiceless mathematics becomes easier.

Answer 2

The axiom of choice is a result of existence (whose statement is in terms of sets and functions) in many books of analysis is used only to prove another result of existence: Zorn's Lemma whose statement is in terms of sets and weak order. See for example Analysis Now by Gert K. Pedersen.

The vast majority of books of functional analysis only enunciate Zorn's lemma without proving it from the axiom of choice. The books of mathematical analysis in $\mathbb{R}^n$ need to state, for their purposes, only weaker results than Zorn's lemma: supremum axiom and Well-ordering principle (Or the best known equivalent of the principle of the well ordering: Principle of Mathematical Induction).

The only elementary book of Mathematical Analysis that I have in mind at the moment which indicates more than three results directly from the axiom of choice is the one already mentioned above.

But most of the functional analysis books, prove several interesting consequences of Zorn's lemma for example:

• $ \mbox{Zorn's lemma}\implies \mbox{Hahn–Banach theorems} \implies \mbox{Separating Hyperplane theorems}\implies\ldots $ $ \quad \ldots\implies \mbox{Banach–Alaoglu theorem} $

• $ \mbox{Zorn's lemma}\implies \mbox{supremum axiom} \implies \mbox{Intermediate value theorem}\implies$ $\implies \mbox{Rolle's theorem} \implies \mbox{Mean value theorem} \implies \mbox{Fundamental theorem of calculus}. $

In the context of non-linear functional analysis: $\mbox{Zorn's lemma}\implies$ Ekeland variational principle.