This text, based on the lectures of a course presented at the University of Coimbra, and written in a very didactic and engaging style, is a careful and innovative exposition of the pointfree theory of rings of continuous real functions and its constructive aspects. 

A global treatment of pointfree continuous real functions put them in a new perspective. Highlights of the book include an in-depth treatment of Stone-Weierstrass theorem and its applications to functional analysis. 

The material appeared here in book form for the first time and became the main reference on the topic, with many citations in MathSciNet and Zentralblatt. Includes supplementary material and three appendices provide the necessary background on concepts from logic, ordered rings and uniform structures.

The book will be of use for researchers already active in the area, but also for those interested in this growing field (sometimes even penetrating into some parts of theoretical computer science), for graduate and PhD students, and others.

Professor Bernhard Banaschewski was born in Munich on March 22, 1926. He studied at the University of Hamburg and received his doctorate in mathematics from Ernst Witt in 1954. The postwar situation in Germany prompted him to move to McMaster University in Hamilton, Ontario, Canada, in 1955. He made rapid progress there and was awarded the prestigious McKay Professorship of Mathematics in 1964. He retired as an emeritus McKay professor in 1991. Since 1962, he was a Fellow of the Royal Society of Canada. His more than 200 publications cover areas such as general topology, pointfree topology, order theory, algebra, category theory, and the foundations of mathematics. He made significant contributions to the development of point-free topology and its applications to various areas of mathematics, including topos theory, mathematical foundations, functional analysis and algebraic topology.