Normed and Banach spaces, examples and applications
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 pl
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n 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. The chapter will, in essence, introduce the working language and the elementary topological instruments of the theory of linear operators. Even if mostly selfcointained, the chapter is by no means exhaustive if compared to the immense literature on the basic properties of normed and Banach spaces. The texts [Rud82, Rud91] should be consulted in this respect. In due course we shall specialise to operators on complex Hilbert spaces, with a short detour in Chapter 4 into the more general features of compact operators. The most important notions of the present chapter are without any doubt bounded operators and the various topologies (induced by norms or seminorms) on spaces of operators. The relevance of these mathematical tools descends from the fact that the language of linear operators on linear spaces is the language used to formulate QM. Here the class of bounded operators plays a central technical part, even though in QM one is forced, on physical grounds, to introduce and work with unbounded operators too, as we shall see in the second part of the book. The chapter’s first part is devoted to the elementary notions of normed space, Banach space and their basic topological properties. We will discuss examples, like the space of continuous maps C(K) over a compact space K, and prove the crucial theorem of Arzelà–Ascoli in this setup. In the examples will also prove key results such as the completeness of L p spaces (Fischer–Riesz theorem). In the second part we will define the norm of an operator and establish its main features. Part three will discuss the fundamental theorems of Banach spaces, in their simplest versions. These are the theorems of Hahn–Banach, Banach–Steinhaus, plus the open mapping theorem a corollary of Baire’s category theorem. We will prove a few useful technical consequences (the inverse operator theorem and the closed graph theorem). Then we will introduce the various operator topologies that come into play, Moretti V.: Spectral Theory and Quantum Mechanics Unitext – La Matematica per il 3+2 DOI 10.1007/978-88-470-2835-7_2, © Springer-Verlag Italia 2013
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2 Normed and Banach spaces, examples and applications
prove the theorem of Banach–Alaoglu and briefly recall the Krein–Milman theorem and Fréchet spaces. Part four will be devoted to projection operators in normed spaces. This we will specialise in the next chapter to that – more useful for our purposes – of an orthogonal projector. In the final two sections we will treat two elementary but important topics: equivalent norms (including a proof that n-dimensional normed spaces are Banach and homeomorphic to the standard Cn ) and the theory of contractions in complete normed spaces (including, in the examples, a proof of the lo
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