Techniques of Constructive Analysis - (Universitext) by Douglas S Bridges & Luminita Simona Vita (Paperback)

About the Book

This book is an introduction to constructive mathematics with an emphasis on techniques and results that have been obtained in the last twenty years. There is specific focus on techniques and themes, and the book includes many simplified or improved proofs.

Book Synopsis

This book is an introduction to constructive mathematics with an emphasis on techniques and results that have been obtained in the last twenty years. It is the only book to deal with developments in Bishop-style constructive analysis over the past 20 years. The text covers fundamental theory of the real line and metric spaces, focusing on locatedness in normed spaces and with associated results about operators and their adjoints on a Hilbert space. There are two appendices to the book. The first gathers together some basic notions about sets and orders; the second gives the axioms for intuitionistic logic. The intended readership of the book consists of senior undergraduate and graduate students, as well as professional research mathematicians. No background in intuitionistic logic or constructive analysis is needed in order to read the book, but some familiarity with the classical theories of metric, normed and Hilbert spaces is necessary.

From the Back Cover

This text provides a rigorous, wide-ranging introduction to modern constructive analysis for anyone with a strong mathematical background who is interested in the challenge of developing mathematics algorithmically. The authors begin by outlining the history of constructive mathematics, and the logic and set theory that are used throughout the book. They then present a new construction of the real numbers, followed by the fundamentals of the constructive theory of metric and normed spaces; the lambda-technique (a special method that enables one to prove many results that appear, at first sight, to be nonconstructive); finite- dimensional and Hilbert spaces; and convexity, separation, and Hahn-Banach theorems. The book ends with a long chapter in which the work of the preceding ones is applied to operator theory and other aspects of functional analysis. Many results and proofs, especially in the later chapters, are of relatively recent origin.

The intended readership includes advanced undergraduates, postgraduates, and professional researchers in mathematics and theoretical computer science. With this book, the authors hope to spread the message that doing mathematics constructively is interesting and challenging, and produces new, deep computational information.

Review Quotes

From the reviews:

"Constructive mathematics is often mistakenly thought to be a branch of formal logic, or to involve recursive function theory. The recent Bridges-Vîta textbook is an exemplar of perseverance in efforts to correct this situation. For students at the graduate level it is an excellent introduction to constructive mathematics: for the more experienced reader it is a portal to some of the latest research using constructive methods." (Mark Mandelkern, Zentralblatt MATH, Vol. 1107 (9), 2007)

"Bridges and Vita (both, Univ. of Canterbury, New Zealand) offer a book, written jointly by an associate (Bridges) of the famous Errett Bishop ... . Thorough index; plentiful exercises and references. Eminently suitable. Summing Up: Recommended. Upper-division undergraduates through faculty." (F. E. J. Linton, CHOICE, Vol. 44 (11), August, 2007)

"The authors have written a valuable introduction to constructive analysis, devoting the larger part of their space and efforts to subjects from constructive functional analysis, a field to which they have made important contributions themselves. ... Without a doubt, this well-written book shows constructive analysis to be a serious and important subject. The authors, who have thought on their subject long and deeply, deserve our thanks." (Wim Veldman, Mathematical Reviews, Issue 2008 a)