For us a topological space is a space where on has given a sense to:
Indeed, the most general notion of limit is expressed in topological
A sequence of points of a topological space has a limit if
each neighbourhood of contains the terms of the sequence after a
Continuity of functionals
The space and its topology
Distances and metrics
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.
Tensors and metrics
If the space has a metrics then variance can be changed
easily. A metrics allows to measure distances between points in the
space. The elementary squared distance between two points and
Covariant components can be expressed with respect to contravariant
The invariant can be written
and tensor like can be written:
Limits in the distribution's sense
Let be a family of distributions depending on a real parameter
. Distribution tends to distribution when tends
In particular, it can be shown that distributions associated to functions verifying:
converge to the Dirac distribution.
Family of functions
over the interval
et zero anywhere else convergesto the Dirac distribution when
tends to zero.
Figure figdirac represents an example of such a family of functions.