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
To each metrical space can be associated a topological space.
In this text, all the topological spaces considered are
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
A normed vectorial space is a vectorial space equipped 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
It is thus a metrical space by using the distance induced by the norm
associated to the scalar
A Hilbert space is a complete separated prehilbertian space.
The space of summable squared functions is a Hilbert space.