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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.

Calculus. --- Mathematical analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Analysis (Mathematics) --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Functions --- Geometry, Infinitesimal --- Logic, Symbolic and mathematical. --- Global analysis (Mathematics). --- Mathematics. --- Functional analysis. --- Operator theory. --- Mathematical Logic and Foundations. --- Analysis. --- Real Functions. --- Functional Analysis. --- Operator Theory. --- Functional analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Math --- Science --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Mathematical logic. --- Analysis (Mathematics). --- Functions of real variables. --- Real variables

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A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community. It is this concept that sets mathematics apart from other disciplines and distinguishes it as the prototype of a deductive science. Proofs thus are utterly relevant for research, teaching and communication in mathematics and of particular interest for the philosophy of mathematics. In computer science, moreover, proofs have proved to be a rich source for already certified algorithms. This book provides the reader with a collection of articles covering relevant current research topics circled around the concept 'proof'. It tries to give due consideration to the depth and breadth of the subject by discussing its philosophical and methodological aspects, addressing foundational issues induced by Hilbert's Programme and the benefits of the arising formal notions of proof, without neglecting reasoning in natural language proofs and applications in computer science such as program extraction.

Proof theory. --- Mathematics. --- Logic, Symbolic and mathematical. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Math --- Science --- Logic, Symbolic and mathematical --- Mathematical Logic. --- Philosophy of Mathematics. --- Theoretical Computer Science.