Summable square

In Mathematical, one says that a measurable function $f$ of $\backslash \; R$ in $\backslash \; mathbb\; \{C\}$ is of square summable when the quantity

$S\; =\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; |F\; (X)|^2\; \backslash \; mathrm\; \{D\}\; x$

is a finished number.

The whole of such functions forms a vector Space, which one can provide with a structure of Espace of Hilbert using the scalar Produit according to

$(F,\; G)\; =\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (X)\; \backslash \; overline\; \{G\}\; (X)\; \backslash \; mathrm\; \{D\}\; x$

It is a very important point in Quantum physics also: if one considers a Fonction of wave $|\backslash \; Psi\; (\backslash \; vec\; R,\; T)\; \backslash \; rangle$ associated with a particle, then, according to the equation of Schrödinger, the quantity S stated Ci above represents the Densité of probability of finding the particle in all space, such a number must be ranging between 0 and 1. Any function of wave is thus of square summable.

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