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Mathematics of Quantization and Quantum Fields by Christian Gérard, Jan Derezinski

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2

Operators in Hilbert spaces

In this chapter we recall basic properties of operators on topological vector spaces. We concentrate on Hilbert spaces, which play the central role in quantum physics.

2.1 Convergence and completeness

We start with a discussion of various topics related to convergence and completeness.

2.1.1 Nets

Nets are generalizations of sequences. In this subsection we briefly recall this useful concept.

Definition 2.1 A directed set is a set I equipped with a partial order relation ≤ such that for any i, j ∈ I there exists k ∈ I such that i ≤ k, j ≤ k.

We will often use the following directed set:

Definition 2.2 Let I be a set. We denote by 2finI the family of finite subsets of I. It becomes a directed set when we equip it ...

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