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.