Lebesgue's lemma
Encyclopedia
For Lebesgue's lemma for open covers of compact spaces in topology see Lebesgue's number lemma
In mathematics
, Lebesgue's lemma is an important statement in approximation theory
. It provides a bound for the projection error.
, U be a subspace of V and let
be a linear projector
on
. Then, for each v in V:
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:Such a number δ is called a Lebesgue number of this cover. The notion of a Lebesgue number itself is useful in other applications as well.-References:-External links:*...
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Lebesgue's lemma is an important statement in approximation theory
Approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby...
. It provides a bound for the projection error.
Statement
Let (V, ||·||) be a normed vector spaceNormed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
, U be a subspace of V and let
be a linear projectorProjection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....
on
. Then, for each v in V: