# Introduction to Mathematical Physics/Topological spaces

## Definition

For us a topological space is a space where one has given a sense to:

${\displaystyle \lim u_{n}=u.}$

Indeed, the most general notion of limit is expressed in topological spaces:

Definition:

A sequence ${\displaystyle u_{n}}$  of points of a topological space has a limit ${\displaystyle l}$  if each neighbourhood of ${\displaystyle l}$  contains the terms of the sequence after a certian rank.

## Distances and metrics

Definition:

A distance on an ensemble ${\displaystyle E}$  is an map ${\displaystyle d}$  from ${\displaystyle E\times E}$  into ${\displaystyle R^{+}}$  that verfies for all ${\displaystyle x,y,z}$  in ${\displaystyle E}$ :

${\displaystyle d(x,y)=0}$  if and only if ${\displaystyle x=y}$ . ${\displaystyle d(x,y)=d(y,x)}$ . ${\displaystyle d(x,z)\leq d(x,y)+d(y,z)}$ .

Definition:

A metrical space is a couple ${\displaystyle (A,d)}$  of an ensemble ${\displaystyle A}$  and a distance ${\displaystyle d}$  on ${\displaystyle A}$ .

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 ${\displaystyle \pi }$  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 ${\displaystyle (A,d)}$  a metrical space. A sequence ${\displaystyle x_{n}}$  of elements of ${\displaystyle A}$  is said a Cauchy sequence if ${\displaystyle \forall \epsilon >0\exists n,\forall (p,q),p>n,q>n,d(x_{p},x_{q})<\epsilon }$ .

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 ${\displaystyle (A,d)}$  is said complete if any Cauchy sequence of elements of ${\displaystyle A}$  converges in ${\displaystyle A}$ .

The space ${\displaystyle R}$  is complete. The space ${\displaystyle Q}$  of the rational number is not complete. Indeed the sequence ${\displaystyle u_{n}=\sum _{k=0}^{n}{\frac {1}{k!}}}$  is a Cauchy sequence but doesn't converge in ${\displaystyle Q}$ . It converges in ${\displaystyle R}$  to ${\displaystyle e}$ , that shows that ${\displaystyle e}$  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 ${\displaystyle E}$  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 ${\displaystyle L^{2}}$  is a Hilbert space.

## Tensors and metrics

If the space ${\displaystyle E}$  has a metrics ${\displaystyle g_{ij}}$  then variance can be changed easily. A metrics allows to measure distances between points in the space. The elementary squared distance between two points ${\displaystyle x_{i}}$  and ${\displaystyle x_{i}+dx_{i}}$  is:

${\displaystyle ds^{2}=g_{ij}dx^{i}dx^{j}=g^{ij}dx_{i}dx_{j}}$

Covariant components ${\displaystyle x_{i}}$  can be expressed with respect to contravariant components:

${\displaystyle x_{i}=g_{ij}x^{j}}$

The invariant ${\displaystyle x^{i}y_{j}}$  can be written

${\displaystyle x^{i}y_{j}=x^{i}g_{ij}y^{j}}$

and tensor like ${\displaystyle a_{i}^{j}}$  can be written:

${\displaystyle a_{i}^{j}=g^{jk}a_{ik}}$

## Limits in the distribution's sense

Definition:

Let ${\displaystyle T_{\alpha }}$  be a family of distributions depending on a real parameter ${\displaystyle \alpha }$ . Distribution ${\displaystyle T_{\alpha }}$  tends to distribution ${\displaystyle T}$  when ${\displaystyle \alpha }$  tends to ${\displaystyle \lambda }$  if:

${\displaystyle \forall \phi \in {\mathcal {D}},\lim _{\alpha \rightarrow \lambda }=}$

In particular, it can be shown that distributions associated to functions ${\displaystyle f_{\alpha }}$  verifying:

${\displaystyle f_{\alpha }(x)\geq 0}$
${\displaystyle \int f_{\alpha }(x)dx=1}$
${\displaystyle \forall a\geq 0,\lim _{\alpha \rightarrow \infty }\int _{|x|>a}f_{\alpha }(x)dx=0}$

converge to the Dirac distribution.

figdirac

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