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:

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


Definition:

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


Continuity of functionals[edit]

The space and its topology[edit]

Distances and metrics[edit]

Definition:

A distance on an ensemble is an map from into that verfies for all in :


if and only if . . .




Definition:

A metrical space is a couple of an ensemble and a distance on .

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 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 metrical space. A sequence of elements of is said a Cauchy sequence if .

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 is said complete if any Cauchy sequence of elements of converges in .

The space is complete. The space of the rational number is not complete. Indeed the sequence is a Cauchy sequence but doesn't converge in . It converges in to , that shows that 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 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 is a Hilbert space.

Tensors and metrics[edit]

If the space has a metrics then variance can be changed easily. A metrics allows to measure distances between points in the space. The elementary squared distance between two points and is:

Covariant components can be expressed with respect to contravariant components:

The invariant can be written

and tensor like can be written:


Limits in the distribution's sense[edit]

Definition:

Let be a family of distributions depending on a real parameter . Distribution tends to distribution when tends to if:

In particular, it can be shown that distributions associated to functions verifying:

converge to the Dirac distribution.

figdirac

Family of functions where is over the interval et zero anywhere else convergesto the Dirac distribution when tends to zero.

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