Abstract
In the book’s first proper chapter, we will discuss the fundamental notions and theorems about normed and Banach spaces. We will introduce certain algebraic structures modelled on natural algebras of operators on Banach spaces. Banach operator algebras play a relevant role in modern formulations of Quantum Mechanics.
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Notes
- 1.
The definition generalises trivially to metric spaces.
- 2.
\({\mathrm{M}}_{\epsilon}^{\left(\delta \right)}=\left\{X\in C\left({J}_{\delta };\mathbb{k}\right)\left|\right|\left|X-{X}_0\right|{\Big|}_{\infty}\le \epsilon \right\}\), where X 0 is here the constant map equal to x 0 on J δ . Thus M (δ) ε is the closure of the open ball of radius ε centred at X 0 inside \(C\left({J}_{\delta };{\mathbb{k}}^n\right)\).
- 3.
This inequality descends from (a + b) ≤ 2 max {a, b}, whose pth power reads (a + b)p ≤ 2p max {a p, b p} ≤ 2p (a p + b p).
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© 2013 Springer-Verlag Italia
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Moretti, V. (2013). Normed and Banach spaces, examples and applications. In: Spectral Theory and Quantum Mechanics. UNITEXT(). Springer, Milano. https://doi.org/10.1007/978-88-470-2835-7_2
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DOI: https://doi.org/10.1007/978-88-470-2835-7_2
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-2834-0
Online ISBN: 978-88-470-2835-7
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