''For Lebesgue's lemma for open covers of compact spaces in topology see

Lebesgue's number lemma In topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces. It states: :If the metric space (X, d) is compact and an open cover of X is given, then there exists a number \delta > 0 such ...

'' In

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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. Introdu ...

of the projection.

Statement

Let be a

normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...

, 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 degenerate triangles, but ...

: 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

References

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

Statement

See also

References