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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Lebesgue's lemma is an important statement in
approximation theory In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characteri ...
. 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 In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Inform ...
of the projection.


Statement

Let be a
normed vector space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
, a subspace of , and a linear projector on . Then for each in : : \, v-Pv\, \leq (1+\, P\, )\inf_\, v-u\, . The proof is a one-line application of the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#T ...
: for any in , by writing as , it follows that :\, v-Pv\, \leq\, v-u\, +\, u-Pu\, +\, P(u-v)\, \leq(1+\, P\, )\, u-v\, where the last inequality uses the fact that together with the definition of the
operator norm In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Inform ...
.


See also

* Lebesgue constants


References

* {{DEFAULTSORT:Lebesgue's Lemma Lemmas in mathematical analysis Approximation theory