Space pseudometric

In Mathematical, a space pseudometric is a particular case of Espace hemimetric checking a relation of Symétrie, which generalizes thus the metric concept of Espace. When a Topologie is generated by a family of pseudometric, the space is called Espace of gauge.

The pseudométriques ones appear naturally in analyzes functional.

Definition

A space pseudometric $\backslash \; left\; (X,\; \backslash \; mathrm\; D\; \backslash \; right)$ is the data of a Ensemble X and of a positive function with actual values $\backslash \; mathrm\; D:\; X\; \backslash \; times\; X\; \backslash \; longrightarrow\; \backslash \; mathbb\; \{R\}$, called pseudometric function (or pseudometric ), which checks the three following relations:

• $\backslash \; forall\; X\; \backslash \; in\; X,\; \backslash \; quad\; \backslash \; mathrm\; D\; \backslash \; left\; (X,\; X\; \backslash \; right)\; =\; 0$;

• $\backslash \; forall\; X,\; there\; \backslash \; in\; X,\; \backslash \; quad\; \backslash \; mathrm\; D\; \backslash \; left\; (X,\; there\; \backslash \; right)\; =\; \backslash \; mathrm\; D\; \backslash \; left\; (there,\; X\; \backslash \; right)$ (Symmetry);
• $\backslash \; forall\; X,\; there,\; Z\; \backslash \; in\; X,\; \backslash \; quad\; \backslash \; mathrm\; D\; \backslash \; left\; (X,\; Z\; \backslash \; right)\; \backslash \; Leq\; \backslash \; mathrm\; D\; \backslash \; left\; (X,\; there\; \backslash \; right)\; +\; \backslash \; mathrm\; D\; \backslash \; left\; (there,\; Z\; \backslash \; right)$ (triangular Inequality).

With the difference of a metric Space, the points of a space pseudometric are not necessarily distinct — i.e. one can have $\backslash \; mathrm\; D\; (X,\; there)\; =0$ for distinct values $x\; \backslash \; y$.

Examples

That is to say space $\backslash \; mathcal\; \{F\}\; \backslash \; left\; (X\; \backslash \; right)$ of the functions with actual values $f:\; X\; \backslash \; to\; \backslash \; mathbb\; \{R\}$, added point $x\_0\; \backslash \; in\; X$. This point induces pseudométrique on the space of the functions, given by:

$\backslash \; forall\; F,\; G\; \backslash \; in\; \backslash \; mathrm\; F\; \backslash \; left\; (X\; \backslash \; right),\; \backslash \; quad\; \backslash \; mathrm\; D\; \backslash \; left\; (F,\; G\; \backslash \; right)\; =\; |\; F\; \backslash \; left\; (x\_0\; \backslash \; right)\; -\; G\; \backslash \; left\; (x\_0\; \backslash \; right)|\backslash ;$

For a vector Space V , a seminorme p induces pseudométrique on V :

$\backslash \; mathrm\; D\; \backslash \; left\; (X,\; there\; \backslash \; right)\; =\; p\; \backslash \; left\; (X\; there\; \backslash \; right)$

Topological properties

The pseudometric Topologie is induced by the whole of the swell S open:

who forms a Base topology. A topological Espace is known as pseudométrisable if one can provide space with a pseudometric topology.

Metric identification

The cancellation of pseudometric the armature a relation of equivalence, called metric identification, which makes space pseudometric a metric Espace complete. That can be made by defining $x\; \backslash \; sim\; y$ if $\backslash \; mathrm\; D\; \backslash \; left\; (X,\; there\; \backslash \; right)\; =0$. That is to say $X^*=X/\backslash \; sim$ and let us pose:

$\backslash \; mathrm\; d^\; \{*\}\; \backslash \; left\; (\backslash \; left,\; \backslash \; left\; \backslash \; right)\; =\; \backslash \; mathrm\; D\; \backslash \; left\; (X,\; there\; \backslash \; right)$

Then $\backslash \; mathrm\; d^\; \{*\}$ is a Métrique on $X^*$ and $\backslash \; left\; (X^\; \{*\},\; \backslash \; mathrm\; d^\; \{*\}\; \backslash \; right)$ is a metric Espace well defined.

The metric identification preserves induced topologies: a subset $A\; \backslash \; subset\; X$ is open ( resp. closed) of $\backslash \; left\; (X,\; \backslash \; mathrm\; D\; \backslash \; right)$ If and only if $\backslash \; pi\; \backslash \; left\; (\backslash \; right\; has)\; =\; \backslash \; left$ is open ( resp. closed) of $\backslash \; left\; (X^\; \{*\},\; \backslash \; mathrm\; d^\; \{*\}\; \backslash \; right)$.

References

Random links:

Marie Susini | Señora Jessica | Frederic Poulon | Lerma (Italy) | Clement Lockquell | Čokotar | Daron_Acemoglu

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