From Euclidean to Hilbert Spaces

Appendix 1Quotient Space

The concept of quotient of a vector space is essential in mathematics, and, in our opinion, does not always receive the attention it deserves in works on linear algebra. For this reason, we have chosen to devote an appendix to the definition and interpretation of this concept.

DEFINITION A1.1.– An equivalence relation ~ defined on a vector space V (of arbitrary dimension) on the field is said to be compatible with the linear structure of V if:

The equivalence class of 0 in V is a vector space Z (since it is stable with respect to linear combinations, and contains the neutral element) known as the kernel of the equivalence relationship.

One special case of this definition is when w = w′ v′ and α = 1, β = –1, which implies:

Conversely, if v – v′ ∈ Z, i.e. v – v^′ ~ 0 v – v^′ ~ v^′ – v^′, and by the fact that v^′ ~ v^′, and since ~ is compatible with the linear structure of V, we obtain: v – v^′ + v^′ ~ v^′ – v^′ + v^′, that is, v ~ v^′.

In short: v ~ v^′ v – v^′ ∈ Z, which ...