Abstract
In this first chapter, we provide the definition and the basic properties of a Hilbert space H, together with some examples of spaces which will be needed in the next chapters.
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Notes
- 1.
Here, z ∗ is the complex conjugate of z (so z ∗ = z if \(z \in \mathbb{R}\)).
- 2.
Here, \(\mathfrak{R}(z)\) denotes the real part of the complex number z.
- 3.
Some authors prefer calling subspace and closed subspace what we have called linear manifold and subspace, respectively.
- 4.
Some authors prefer to say Hilbert basis.
Bibliography
H. Brezis, Analyse Fonctionnelle, Masson, Paris, 1983.
A. Fonda, On a geometrical formula involving medians and bimedians, Math. Mag. 86 (2013), 351–357.
A. Fonda, A generalization of the parallelogram law to higher dimensions, preprint.
G. Helmberg, Introduction to Spectral Theory in Hilbert Space, North-Holland, Amsterdam, 1969.
T.W. Körner, Fourier Analysis, Cambridge University Press, Cambridge, 1989.
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Fonda, A. (2016). Preliminaries on Hilbert Spaces. In: Playing Around Resonance. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-47090-0_1
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DOI: https://doi.org/10.1007/978-3-319-47090-0_1
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