Introduction to Mathematical Physics/Topological spaces

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Definition[edit]

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.


Continuity of functionals[edit]

The space {\mathcal D} and its topology[edit]

Distances and metrics[edit]

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[edit]

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[edit]

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 is1/\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.