# Introduction to Mathematical Physics/Topological spaces

## Definition

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

$\lim u_n=u.$

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

Definition:

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

## Distances and metrics

Definition:

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

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

Definition:

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

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

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

The space $R$ is complete. The space $Q$ of the rational number is not complete. Indeed the sequence $u_n=\sum_{k=0}^n\frac{1}{k!}$ is a Cauchy sequence but doesn't converge in $Q$. It converges in $R$ to $e$, that shows that $e$ is irrational.

Definition:

A normed vectorial space is a vectorial space equiped 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 $E$ 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 $L^2$ is a Hilbert space.

## Tensors and metrics

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

$ds^2=g_{ij}dx^idx^j=g^{ij}dx_idx_j$

Covariant components $x_i$ can be expressed with respect to contravariant components:

$x_i=g_{ij}x^j$

The invariant $x^iy_j$ can be written

$x^iy_j=x^ig_{ij}y^j$

and tensor like $a_i^j$ can be written:

$a_i^j=g^{jk}a_{ik}$

## Limits in the distribution's sense

Definition:

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

$\forall \phi \in {\mathcal D}, \lim_{\alpha\rightarrow\lambda}=$

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

$f_\alpha(x)\geq 0$

$\int f_\alpha(x)dx=1$

$\forall a\geq 0, \lim_{\alpha\rightarrow\infty}\int_{|x|>a}f_\alpha(x)dx=0$

converge to the Dirac distribution.

figdirac

Family of functions $f_\epsilon$ where $f_\epsilon$ is$1/\epsilon$ over the interval $[0,\epsilon]$ et zero anywhere else convergesto the Dirac distribution when $\epsilon$ tends to zero.

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