Lebesgue's lemma
{{For|Lebesgue's lemma for open covers of compact spaces in topology|Lebesgue's number lemma}}
In mathematics, Lebesgue's lemma is an important statement in approximation theory. It provides a bound for the projection error, controlling the error of approximation by a linear subspace based on a linear projection relative to the optimal error together with the operator norm of the projection.
Statement
Let {{math|(V, {{!!}}·{{!!}})}} be a normed vector space, {{mvar|U}} a subspace of {{mvar|V}}, and {{mvar|P}} a linear projector on {{mvar|U}}. Then for each {{mvar|v}} in {{mvar|V}}:
:
The proof is a one-line application of the triangle inequality: for any {{mvar|u}} in {{mvar|U}}, by writing {{math|v − Pv}} as {{math|(v − u) + (u − Pu) + P(u − v)}}, it follows that
:
where the last inequality uses the fact that {{math|u {{=}} Pu}} together with the definition of the operator norm {{math|{{!!}}P{{!!}}}}.
See also
References
- {{cite book|mr=1261635|last1=DeVore|first1=Ronald A.|last2=Lorentz|first2=George G.|title=Constructive approximation|series=Grundlehren der mathematischen Wissenschaften|volume=303|publisher=Springer-Verlag|location=Berlin|year=1993|isbn=3-540-50627-6|zbl=0797.41016|author-link1=Ronald DeVore|author-link2=George G. Lorentz}}
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