This classic work by the late Stefan Banach has been translated into English so as to reach a yet wider audience. It contains the basics of the algebra of operators, concentrating on the study of linear operators, which corresponds to that of the linear forms a1x1 + a2x2 + ... + anxn of algebra. The book gathers results concerning linear operators defined in general spaces of a certain kind, principally in Banach spaces, examples of which are: the space of continuous functions, that of the pth-power-summable functions, Hilbert space, etc. The general theorems are interpreted in various mathematical areas, such as group theory, differential equations, integral equations, equations with infinitely many unknowns, functions of a real variable, summation methods and orthogonal series. A new fifty-page section (``Some Aspects of the Present Theory of Banach Spaces'') complements this important monograph.

Table of Contents

The Lebesgue-Stieltjes Integral. (B)-Measurable Sets and Operators in Metric Spaces. Groups. General Vector Spaces. F-Spaces. Normed Spaces. Banach Spaces. Compact Operators. Biorthogonal Sequences. Linear Functionals. Weakly Convergent Sequences. Linear Functional Equations. Isometry, Equivalence, Isomorphism. Linear Dimension. Appendix: Weak Convergence in Banach Spaces. Some Aspects of the Present Theory of Banach Spaces by A. Pelczyński and Cz. Bessaga. Reflexive and Weakly Compactly Generated Banach Spaces. Related Counter Examples. Local Properties of Banach Spaces. The Approximation Property and Bases. Characterizations of Hilbert Spaces in the Class of Banach Spaces. Classical Banach Spaces. The Topological Structure of Linear Metric Spaces. Bibliography.


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© 1987
North Holland
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