Introduction to Mathematical Physics/Topological spaces

Definition

For us a topological space is a space where one has given a sense to: Template:IMP/eq Indeed, the most general notion of limit is expressed in topological spaces:

Distances and metrics

Template:IMP/defn 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. Template:IMP/defn 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. Template:IMP/defn 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.

Template:IMP/defn The norm induced a distance, so a normed vectorial space is a topological space (on can speak about limits of sequences).

Template:IMP/defn It is thus a metrical space by using the distance induced by the norm associated to the scalar product.

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 $x_i+d{x}_i$ is: Template:IMP/eq

Covariant components $x_i$ can be expressed with respect to contravariant components: Template:IMP/eq The invariant $x^i{y}_j$ can be written Template:IMP/eq and tensor like $a_i^j$ can be written: Template:IMP/eq

Limits in the distribution's sense

Template:IMP/defn In particular, it can be shown that distributions associated to functions $f_{\alpha }$ verifying: Template:IMP/eq Template:IMP/eq Template:IMP/eq converge to the Dirac distribution. Template:IMP/label

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

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