Introduction to Mathematical Physics/Topological spaces

Definition

For us a topological space is a space where one 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 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 $E$ 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:

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

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

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

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

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

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:

$\forall \phi \in {\mathcal {D}},\lim _{\alpha \rightarrow \lambda }<T_{\alpha },\phi >=<T,\phi >$

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.