# Introduction to Mathematical Physics/Topological spaces

## Contents

## Definition[edit]

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 functionals[edit]

### The space and its topology[edit]

## Distances and metrics[edit]

**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 metrics[edit]

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[edit]

**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

Figure figdirac represents an example of such a family of functions.