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Hilbert Spaces
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
Hilbert spaces are the most important examples of Banach spaces. Their study was initiated by Hilbert’s results at the beginning of the twentieth century; his results are related to the spaces ℓ2 and L2. Banach spaces have some properties of the Euclidean space Cn (Theorem 3.2.7), while some geometric properties of Euclidean spaces such as, for instance, orthogonality, also exist in Hilbert spaces. Furthermore, the study of Hilbert spaces is connected with the operator theory on Hilbert spaces. Hence we also study operators on Hilbert spaces in this chapter.
Combined Learning
Published in Hamidou Tembine, Distributed Strategic Learning for Wireless Engineers, 2018
Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are required to be complete (if every Cauchy sequence of points in X has a limit that is also in X), a property that stipulates the existence of enough limits in the space to allow the techniques of calculus to be used.
Signal Space
Published in Jerry D. Gibson, Mobile Communications Handbook, 2017
The concept of signal space has its roots in the mathematical theory of inner product spaces known as Hilbert spaces (Stakgold, 1967). Many books on linear systems touch on the subject of signal spaces in the context of Fourier series and transforms (Ziemer et al., 1998). The applications of signal space concepts in communication theory find their power in the representation of signal detection and estimation problems in geometrical terms, which provides much insight into signaling techniques and communication system design. The first person to have apparently exploited the power of signal space concept in communication theory was the Russian Kotel'nikov (1968), who presented his doctoral dissertation in January, 1947. Wozencraft and Jacobs (1965) expanded on this approach and their work is still today widely referenced. Arthurs and Dym (1962) made use of signal space concepts in the performance analysis of several digital modulation schemes. A one-chapter summary of the use of signal space methods in signal detection and estimation is provided in Ziemer and Tranter (2009). Another application of signal space concepts is in signal and image compression. Wavelet theory (Rioul and Vetterli, 1991) is currently finding use in these application areas. Finally, the application of signal space concepts to nonlinear filtering is discussed. In the next section, the fundamentals of generalized vector spaces are summarized, followed by an overview of several applications to signal representations.
Local well-posedness and small data scattering for energy super-critical nonlinear wave equations
Published in Applicable Analysis, 2021
We write or to indicate for some constant C>0. The notation denotes for any small ϵ, and also for . Denote and . The Hilbert space is a Banach space of elements such that , where denotes the Fourier transform , and equipped with the norm . The critical case for is , and equipped with the norm . An usual property of the Fourier transform is the Plancherel equality, that is, . We also have an embedding theorem that for any , . Throughout the whole paper, the letter C will denote various positive constants which are of no importance in our analysis. We use the following norms to denote the mixed spaces , that is,