Tools of the Trade

If you are a beginner or a person who wants to self study some basic advanced math, the book might not be for you. However, if there is someone who knows math really well and who can help you when you get confused, you will find that this is a wonderful book. It has got a nice structure and the proofs, though not easily understood sometimes, are very succinct. In short, a great book for instructors.

in very good condition

This book does not contribute to your enjoyment of mathematics. It has no complete proofs, as Sally's idea of a complete proof is to inform the reader that "this is an interesting proof, you should work it out yourself". I have had multiple classes where the professors stated that, although they were required to list the text, as required for the course, that they would not reference it and that they would have self-contained lectures. But don't worry! Lest you think that this book is not worth buying, be aware that its form factor is quite small. This, combined with its hard covers, make it an excellent book for using as a pad for writing on.

For learning introductory analysis, this is the gold standard--there is no better text. Sally's book covers both the basics (from elementary set theory through metric space topology, with linear algebra and the construction of the reals along the way) and atypical/enriching projects on related content, all in an extremely clear and invitingly casual presentation. It is true that Sally leaves some proofs as exercises, but nothing mission-critical. And after all, mathematics is learnt by doing, not by head-nodding: this is a book for students of mathematics, not an encyclopedia for those who've forgotten a step in a proof. As such, it contains many excellent exercises and investigations, in which Sally weaves in algebra, topology, and logic with the main theme of analysis in a natural way that reinforces the student's learning, stresses important connections across the main branches of mathematics, and ideally prepares the student for future study. The book is also remarkably well-polished. It contains very few typos (and none that cause any lack of clarity)--I think I've spotted only two or three after multiple times teaching from the text. It also features the most intuitive (and honestly the only logical) chapter-section-statement numbering scheme of any mathematics text, wherein theorems, definitions, etc. all increment sequentially with the chapter and section, so there's none of the nonsense you get in texts written by less thoughtful authors, i.e. where the reader ends up in a futile search for Theorem 6.3, which comes after Lemma 6.4, and before Definition 6.2....all in Section 6.5. In short, this is the right book to transition a beginning student of mathematics into one capable of picking up any other undergraduate text and making headway. The reader who finishes it should just prepare to be disappointed when picking up the next book in her mathematical odyssey, as odds are it will be written by a different author neither as clear nor as enriching as Sally.

Mr. Sally's book starts with basic, almost obvious elements of the set theory that provide the logical foundation of essentially all modern mathematics. In this way, he mines the best of the 20th century's progress in solidifying those foundations. But fear not, he does not become mired in Russelian logic. Instead, he goes on to derive and define the properties of myriad basic mathematics, up through some of the basics of topology, complex analysis and so on. A variety of statements are left for the reader to prove, though all important theorems at this level are at least stated explicitly. The reader will go through essentially the same mental development undergone by students fortunate enough to take the famous U of Chicago Math 207 course. Budding mathematicians have been inspired by that course for generations. The proofs, where provided, are pithy and crystal clear while the exercises have a fine distribution of difficulty from trivial to tricky. This is not a reference text by any means, but rather, as the title implies, a means to give you the skill set necessary to understand and invent real mathematics.

Uno vede come sottotitolo di questo libro la frase "Introduction to Advanced Mathematics" e magari si preoccupa un po'. Preoccupazione che si accresce quando legge sulla quarta di copertina che (traduzione mia) "Il libro fornisce una transizione dagli aspetti basati sulle formule dei primi studi universitari di matematica al mondo ricco e creativo degli argomenti più avanzati". Avrei pensato quindi a materiale dell'inizio della laurea magistrale, o del secondo biennio ai lontani tempi in cui matematica la studiavo io e con quindici esami uno si laureava; invece dei cinque capitoli del libro i primi quattro seguivano esattamente l'approccio delle mie vecchie lezioni da matricola. Sarà che a Pisa sono più teorici, ma direi che l'università italiana di trent'anni fa era ben più avanti di quella americana attuale... Il materiale è ben presentato e può tranquillamente essere usato per un ripasso o per studiare da soli, nel caso si sia interessati a questi temi; ma resta comunque un semplice manuale universitario, anche nell'ultima parte sui numeri p-adici (che tra l'altro sarebbe stata più interessante se alla fine fosse stato mostrato un modello visibile dei numeri, in modo da mischiare la teoria ala pratica).