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- W1489213359 abstract "After the introduction to frames in finite-dimensional vector spaces in Chapter 1, the rest of the book will deal with expansions in infinite-dimensional vector spaces. Here great care is needed: we need to replace finite sequences ({f_{k}}_{k=1}^{n}) by infinite sequences ({f_{k}}_{k=1}^{infty }), and suddenly the question of convergence properties becomes a central issue. The vector space itself might also cause problems, e.g., in the sense that Cauchy sequences might not be convergent. We expect the reader to have a basic knowledge about these problems and the way to circumvent them, but for completeness we repeat the central definitions and results concerning Banach spaces and operators hereon in Sections 2.1–2.2. In Sections 2.3–2.4 we specialize to Hilbert spaces and their operators. Section 2.5 deals with pseudo-inverse operators; this subject is not expected to be known and is treated in more detail. Section 2.6 introduces the so-called moment problems in Hilbert spaces. In Sections 2.7–2.9, we discuss the Hilbert space (L^{2}(mathbb{R})) consisting of the square integrable functions on (mathbb{R}) and three classes of operators hereon, as well as the Fourier transform. The material in those sections is not needed for the study of frames and bases on abstract Hilbert spaces, but it forms the basis for all the constructions in Chapters 9– 20" @default.
- W1489213359 created "2016-06-24" @default.
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- W1489213359 date "2016-01-01" @default.
- W1489213359 modified "2023-09-27" @default.
- W1489213359 title "Infinite-Dimensional Vector Spaces and Sequences" @default.
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- W1489213359 doi "https://doi.org/10.1007/978-3-319-25613-9_2" @default.
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