K. Parthasarathy - Böcker

This book starts with a discussion of the classical intermediate value theorem and some of its uncommon “topological” consequences as an appetizer to whet the interest of the reader. It is a concise introduction to topology with a tinge of historical perspective, as the author’s perception is that learning mathematics should be spiced up with a dash of historical development. All the basics of general topology that a student of mathematics would need are discussed, and glimpses of the beginnings of algebraic and combinatorial methods in topology are provided.All the standard material on basic set topology is presented, with the treatment being sometimes new. This is followed by some of the classical, important topological results on Euclidean spaces (the higher-dimensional intermediate value theorem of Poincaré–Miranda, Brouwer’s fixed-point theorem, the no-retract theorem, theorems on invariance of domain and dimension, Borsuk’s antipodal theorem, the Borsuk–Ulam theorem andthe Lusternik–Schnirelmann–Borsuk theorem), all proved by combinatorial methods. This material is not usually found in introductory books on topology. The book concludes with an introduction to homotopy, fundamental groups and covering spaces.Throughout, original formulations of concepts and major results are provided, along with English translations. Brief accounts of historical developments and biographical sketches of the dramatis personae are provided. Problem solving being an indispensable process of learning, plenty of exercises are provided to hone the reader's mathematical skills. The book would be suitable for a first course in topology and also as a source for self-study for someone desirous of learning the subject. Familiarity with elementary real analysis and some felicity with the language of set theory and abstract mathematical reasoning would be adequate prerequisites for an intelligent study of the book.

This core textbook on functional analysis is intended for senior undergraduates and graduate mathematics students. It is suitable for both classrooms and for self-study. The first in a two-volume series presents all the basic material needed for a solid foundation in the subject. It opens with a concise overview of the historical evolution of the subject and goes on to present foundational material in a clear, succinct manner, integrating original source quotes to enrich the narrative and blending the historical perspectives harmoniously with the flow of the subject. Pedagogically, the short chapters are more conducive for learning, and each chapter concludes with applications (including some unusual ones) to diverse fields and exercises to hone students’ understanding. The applications can also serve as sources for student seminars. Various formulations of the spectral theorem and their equivalence are discussed. Different approaches to some important results are presented to enrich the toolkit of the students. The style is neither terse nor verbose, requiring occasional paper-pencil work from the reader. The bibliography is rich and includes all the original works of the founding fathers. Thumbnail biographies of the mathematicians involved should pep up the readers.