By definition, an indefinite inner product space is a real or complex vector space together with a symmetric (in the complex case: hermi tian) bilinear form prescribed on it so that the corresponding quadratic form assumes both positive and negative values. The most important special case arises when a Hilbert space is considered as an orthogonal direct sum of two subspaces, one equipped with the original inner prod uct, and the other with -1 times the original inner product. The subject first appeared thirty years ago in a paper of Dirac [1] on quantum field theory (d. also Pauli [lJ). Soon afterwards, Pontrja gin [1] gave the first mathematical treatment of an indefinite inner prod uct space. Pontrjagin was unaware of the investigations of Dirac and Pauli; on the other hand, he was inspired by a work of Sobolev [lJ, unpublished up to 1960, concerning a problem of mechanics. The attempts of Dirac and Pauli to apply the concept and elemen tary properties of indefinite inner product spaces to field theory have been renewed by several authors. At present it is not easy to judge which of their results will contribute to the final form of this part of physics. The following list of references should serve as a guide to the extensive literature: Bleuler [1], Gupta [lJ, Kallen and Pauli [lJ, Heisen berg [lJ-[4J, Bogoljubov, Medvedev and Polivanov [lJ, K.L.Nagy [lJ-[3], Berezin [lJ, Arons, Han and Sudarshan [1], Lee and Wick [1J.

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I. Inner Product Spaces without Topology.- 1. Vector Spaces.- 2. Inner Products.- 3. Orthogonality.- 4. Isotropic Vectors.- 5. Maximal Non-degenerate Subspaces.- 6. Maximal Semi-definite Subspaces.- 7. Maximal Neutral Subspaces.- S. Projections of Vectors on Subspaces.- 9. Ortho-complemented Subspaces.- 10. Dual Pairs of Subspaces.- 11. Fundamental Decompositions.- Notes to Chapter I.- II. Linear Operators in Inner Product Spaces without Topology.- 1. Linear Operators in Vector Spaces.- 2. Isometric Operators.- 3. Symmetric Operators.- 4. Cayley Transformations.- 5. Principal Vectors of Cayley Transforms.- 6. Pairs of Inner Products: Semi-boundedness.- 7. Pairs of Inner Products: Sign.- 8. Plus-operators.- 9. Pesonen Operators.- 10. Fundamental Projectors.- 11. Fundamental Symmetries. Angular Operators.- Notes to Chapter II.- III. Partial Majorants and Admissible Topologies on Inner Product Spaces.- 1. Locally Convex Topologies on Vector Spaces.- 2. Partial Majorants. The Weak Topology.- 3. Metrizable Partial Majorants.- 4. The Polar of a Normed Partial Majorant.- 5. Admissible Topologies.- 6. Orthogonal Companions and Admissible Topologies.- 7. Projections and Admissible Topologies.- 8. Intrinsic Topology.- 9. Projections and Intrinsic Topology.- Notes to Chapter III.- IV. Majorant Topologies on Inner Product Spaces.- 1. Majorants.- 2. Majorants and Metrizable Partial Majorants.- 3. Orthonormal Systems.- 4. Minimal Majorants.- 5. Majorants and Decomposability.- 6. Decomposition Majorants.- 7. Invariant Properties of E+ and E-.- 8. Subspaces of Spaces with a Hilbert Majorant.- Notes to Chapter IV.- V. The Geometry of Krein Spaces.- 1. Krein Spaces.- 2. Krein Spaces as Completions.- 3. Subspaces.- 4. Maximal Semi-definite Subspaces.- 5. Uniformly Definite Subspaces.- 6.Non-uniformly Definite Subspaces.- 7. Maximal Uniformly Definite Subspaces.- 8. Regular and Singular Subspaces.- 9. Alternating Pairs.- 10. Dissipative Operators in Hilbert Space.- Notes to Chapter V.- VI. Unitary and Selfadjoint Operators in Krein Spaces.- 1. Preliminaries.- 2. The Adjoint of an Operator.- 3. Isometric Operators.- 4. Unitary and Rectangular Isometric Operators.- 5. Spectral Properties of Unitary Operators.- 6. Selfadjoint Operators.- 7. Cayley Transformations.- 8. Unitary Dilations.- Notes to Chapter VI.- VII. Positive Operators and Plus-operators in Krein Spaces.- 1. Positive Operators.- 2. Operators of the Form T*T.- 3. Uniformly Positive Operators.- 4. Plus-operators.- 5. Strict Plus-operators.- 6. Doubly Strict Plus-operators.- Notes to Chapter VII.- VIII. Invariant Semi-definite Subspaces of Linear Operators in Krein Spaces.- 1. Fundamentally Reducible Operators.- 2. Invariant Positive Subspaces of Plus-operators.- 3. Invariant Semi-definite Subspaces of Unitary and Selfadjoint Operators.- 4. Quadratic Pencils of Operators in Hilbert Space.- 5. Quadratic Operator Equations in I-Iilbert Space.- 6. Spectral Functions.- Notes to Chapter VIII.- IX. Pontrjagin Spaces and Their Linear Operators.- 1. The Spaces ?k· Positive Subspaces.- 2. Closed Subspaces.- 3. Isometric Operators: Continuity.- 4. Isometric and Symmetric Operators: Number and Length of Jordan Chains.- 5. Proof of Theorem 4.3.- 6. Regular Symmetric Extensions.- 7. Invariant Positive Subspaces: Existence.- 8. Invariant Positive Subspaces: Uniqueness.- 9. Common Invariant Positive Subspaces for Commuting Operators.- Notes to Chapter IX.- Index of Terms.- Index of Symbols.