Book contents
- Frontmatter
- Contents
- Introduction
- 1 Vector spaces
- 2 Operators in Hilbert spaces
- 3 Tensor algebras
- 4 Analysisin L2 (ℝd)
- 5 Measures
- 6 Algebras
- 7 Anti-symmetric calculus
- 8 Canonical commutation relations
- 9 CCR on Fock space
- 10 Symplectic invariance of CCR in finite-dimensions
- 11 Symplectic invariance of the CCR on Fock spaces
- 12 Canonical anti-commutation relations
- 13 CAR on Fock spaces
- 14 Orthogonal invariance of CAR algebras
- 15 Clifford relations
- 16 Orthogonal invariance of the CAR on Fock spaces
- 17 Quasi-free states
- 18 Dynamics of quantum fields
- 19 Quantum fields on space-time
- 20 Diagrammatics
- 21 Euclidean approach for bosons
- 22 Interacting bosonic fields
- References
- Symbols index
- Subject index
2 - Operators in Hilbert spaces
- Frontmatter
- Contents
- Introduction
- 1 Vector spaces
- 2 Operators in Hilbert spaces
- 3 Tensor algebras
- 4 Analysisin L2 (ℝd)
- 5 Measures
- 6 Algebras
- 7 Anti-symmetric calculus
- 8 Canonical commutation relations
- 9 CCR on Fock space
- 10 Symplectic invariance of CCR in finite-dimensions
- 11 Symplectic invariance of the CCR on Fock spaces
- 12 Canonical anti-commutation relations
- 13 CAR on Fock spaces
- 14 Orthogonal invariance of CAR algebras
- 15 Clifford relations
- 16 Orthogonal invariance of the CAR on Fock spaces
- 17 Quasi-free states
- 18 Dynamics of quantum fields
- 19 Quantum fields on space-time
- 20 Diagrammatics
- 21 Euclidean approach for bosons
- 22 Interacting bosonic fields
- References
- Symbols index
- Subject index
Summary
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.
Convergence and completeness
We start with a discussion of various topics related to convergence and completeness.
Nets
Nets are generalizations of sequences. In this subsection we briefly recall this useful concept.
Definition 2.1A directed set is a set I equipped with a partial order relation ≤ such that for any i, j є I there exists k є I such that i ≤ k, j ≤ k.
We will often use the following directed set:
Definition 2.2Let I be a set. We denote by 2I fin the family of finite subsets of I. It becomes a directed set when we equip it with the inclusion.
Definition 2.3Let S be a set. A net in S is a mapping from a directed set I to S, denoted by ﹛xi﹜iєI.
Definition 2.4A net ﹛xi﹜iєI. in a topological space S converges to x є S if for any neighborhood N of x there exists i є I such that if i ≤ j then xj є N. We will write xi → x. If S is Hausdorff, then a net in S can have at most one limit and one can also write lim xi = x.
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- Information
- Mathematics of Quantization and Quantum Fields , pp. 36 - 56Publisher: Cambridge University PressPrint publication year: 2013