By Steven G. Krantz, Harold R. Parks
This booklet treats the topic of analytic services of 1 or extra genuine variables utilizing, nearly completely, the concepts of genuine research. This process dramatically alters the average development of principles and brings formerly ignored arguments to the fore. the 1st bankruptcy calls for just a history in calculus; the remedy is sort of self-contained. because the publication progresses, the reader is brought to extra subtle subject matters requiring extra historical past and perseverance. while really complicated subject matters are reached, the ebook shifts to a extra expository mode, with targets of introducing the reader to the theorems, offering context and examples, and indicating assets within the literature.
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Extra resources for A Primer of Real Analytic Functions (Birkhauser Advanced Textbooks)
Thus, the structure of Hilbert spaces is more simple and beautiful, and hence, a large number of problems in mathematics, science, and engineering can be successfully treated with geometric methods in Hilbert spaces. This chapter deals with normed spaces, the Lp spaces, generalized functions (distributions), inner product spaces (also called pre-Hilbert spaces), and Hilbert spaces. The fundamental ideas and results are discussed with special attention given to orthonormal systems, linear functionals, and the Riesz representation theorem.
14. Linear operators on a Hilbert space are widely used to represent physical quantities in applied mathematics and physics. In signal processing and wavelet analysis, almost all algorithms are essentially based on linear operators. The most important operators include differential, integral, and matrix operators. In Sect. 15, special attention is given to different kinds of operators and their basic properties. The eigenvalues and eigenvectors are discussed in Sect. 16. Included are several spectral theorems for self-adjoint compact operators and other related results.
In fact, the geometry of Hilbert spaces is more or less a generalization of Euclidean geometry to infinite dimensional spaces. The main reason for this simplicity is that the concept of orthogonality can be introduced in any inner product space so that the familiar Pythagorean formula holds. Thus, the structure of Hilbert spaces is more simple and beautiful, and hence, a large number of problems in mathematics, science, and engineering can be successfully treated with geometric methods in Hilbert spaces.