Introduction to Mathematical Physics/Topological spaces

DefinitionEdit

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:


Definition:

A sequence   of points of a topological space has a limit   if each neighbourhood of   contains the terms of the sequence after a certian rank.


Continuity of functionalsEdit

The space   and its topologyEdit

Distances and metricsEdit

Definition:

A distance on an ensemble   is an map   from   into   that verfies for all   in  :


  if and only if  .  .  .




Definition:

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.

Definition:

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.

Definition:

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.

Definition:

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).

Definition:

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.

Definition:

A Hilbert space is a complete separated prehilbertian space.

The space of summable squared functions   is a Hilbert space.

Tensors and metricsEdit

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 senseEdit

Definition:

Let   be a family of distributions depending on a real parameter  . Distribution   tends to distribution   when   tends to   if:

 

In particular, it can be shown that distributions associated to functions   verifying:

 

 

 

converge to the Dirac distribution.

figdirac

 
Family of functions   where   is  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.