Introduction to Mathematical Physics/Topological spaces
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 spaces:
Continuity of functionals
The space and its topology
Distances and metrics
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.
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.
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.
The norm induced a distance, so a normed vectorial space is a topological space (on can speak about limits of sequences).
It is thus a metrical space by using the distance induced by the norm associated to the scalar product.
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 is:
Covariant components can be expressed with respect to contravariant components:
The invariant can be written
and tensor like can be written:
Limits in the distribution's sense
In particular, it can be shown that distributions associated to functions verifying:
converge to the Dirac distribution.
Figure figdirac represents an example of such a family of functions.