1 - Linear spaces

Summary

In this chapter we recall briefly some salient facts about linear spaces and linear maps. Proofs for the most part are omitted.

Maps

Let X and Y be sets and f : X → Y a map. Then, for each x ∈ X an element f(x) ∈ Y is defined, the subset of Y consisting of all such elements being called the image of f, denoted by im f. More generally f : X ↣ Y will denote a map of an unspecified subset of X to Y, X being called the source of the map and the subset of X consisting of those points x ∈ X for which f(x) is defined being called the domain of f, denoted by dom f. In either case the set Y is the target of f.

Given a map f : X ↣ Y and a point y ∈ Y, the subset f⁻¹{y} of X consisting of those points x ∈ X such that f(x) = y is called the fibre of f over y, this being non-null if and only if y ∈ im f. The set of non-null fibres of f is called the coimage of f and the map

dom f → coim f ; x ↦ f⁻¹{f(x)}

the partition of dom f induced by f. The fibres of a map f are sometimes called the level sets or the contours of f, especially when the target of f is the field of real numbers R.