A distance on an ensemble is an map from into that verfies for all in :
if and only if . . .
A metrical space is a couple of an ensemble and a distance on .
To each metrical space can be associated a topological space. In this text, all the topological spaces considered are metrical space. In a metrical space, a converging sequence admits only one limit (the toplogy is separated ).
Cauchy sequences have been introduced in mathematics when is has been necessary to evaluate by successive approximations numbers like that aren't solution of any equation with inmteger coeficient and more generally, when one asked if a sequence of numbers that are ``getting closer do converge.
Let a metrical space. A sequence of elements of is said a Cauchy sequence if .
Any convergent sequence is a Cauchy sequence. The reverse is false in general. Indeed, there exist spaces for wich there exist Cauchy sequences that don't converge.
A metrical space is said complete if any Cauchy sequence of elements of converges in .
The space is complete. The space of the rational number is not complete. Indeed the sequence is a Cauchy sequence but doesn't converge in . It converges in to , that shows that is irrational.
A normed vectorial space is a vectorial space equiped with a norm.
The norm induced a distance, so a normed vectorial space is a topological space (on can speak about limits of sequences).
A separated prehilbertian space is a vectorial space that has a scalar product.
It is thus a metrical space by using the distance induced by the norm associated to the scalar product.
A Hilbert space is a complete separated prehilbertian space.
The space of summable squared functions is a Hilbert space.