Beppo Levi's inequality

The inequality of Beppo Levi is a result of functional analysis , a branch of mathematics . The inequality goes back to the Italian mathematician Beppo Levi (1875–1961) and is closely linked to the famous projection theorem .

formulation

Given a pre-Hilbert space , provided with the from the underlying scalar derived standard . Furthermore, a subspace and three vectors are given as well . Is now ${\displaystyle {\mathcal {H}}}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle {\mathcal {M}}\subseteq {\mathcal {H}}}$ ${\displaystyle x\in {\mathcal {H}}}$ ${\displaystyle y_{1},y_{2}\in {\mathcal {M}}}$

${\displaystyle d=\inf _{y\in {\mathcal {M}}}\|x-y\|}$

the distance from to , then: ${\displaystyle x}$ ${\displaystyle {\mathcal {M}}}$

${\displaystyle \|y_{1}-y_{2}\|\leq {\sqrt {{\|x-y_{1}\|}^{2}-d^{2}}}+{\sqrt {{\|x-y_{2}\|}^{2}-d^{2}}}}$.

Remarks

• The proof of the inequality is based on conclusions similar to those of the proof of the Cauchy-Bunjakowski-Schwarz inequality .
• The inequality holds especially for every Hilbert space . In the proof of the projection theorem, it delivers the decisive argument, according to which the associated projection operator always exists for a sub-Helbert space . ${\displaystyle {\mathcal {H}}}$
• In the case is and one obtains the triangle inequality . ${\displaystyle {\mathcal {M}}={\mathcal {H}}}$ ${\displaystyle d=0}$

literature

• Mark Neumark : Normalized Algebras . Verlag Harri Deutsch , Thun and Frankfurt / Main 1990, ISBN 3-8171-1001-4 ( MR1038909 ).

Individual evidence

1. ^ Mark Neumark: Normalized Algebras. Verlag Harri Deutsch, Thun and Frankfurt am Main 1990, p. 107 ff.
2. ↑ Neumark, op. Cit., Pp. 108-109.