proof of existence and uniqueness of best approximations

- Proof of the theorem on existence and uniqueness of - ( entry (

Existence : Without loss of generality we can suppose x=0 (we could simply translate by -x the set A).

Let d=d(0,A)=inf{a:aA} be the distance of A to the origin. By defintion of infimumMathworldPlanetmath there exists a sequence (an) in A such that


Let us see that (an) is a Cauchy sequencePlanetmathPlanetmath. By the parallelogram law we have




As A is convex, an+am2A, and therefore


So we see that

an-am2212an2+12am2-d20   whenm,n

which means that an-am0 when m,n, i.e. (an) is a Cauchy sequence.

Since A is completePlanetmathPlanetmathPlanetmathPlanetmath (, ana0 for some a0A.

As a0A its norm must be a0d. But also


which shows that a0=d. We have thus proven the existence of best approximations (

Uniqueness : Suppose there were a0,b0A such that a0=b0=d. Then, by the parallelogram law


If a0-b00 then we would have a0+b022<d2, which is contradictionMathworldPlanetmathPlanetmath since a0+b02A (A is convex).

Therefore a0=b0, which proves the uniqueness of the .

Title proof of existence and uniqueness of best approximations
Canonical name ProofOfExistenceAndUniquenessOfBestApproximations
Date of creation 2013-03-22 17:32:22
Last modified on 2013-03-22 17:32:22
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 6
Author asteroid (17536)
Entry type Proof
Classification msc 49J27
Classification msc 46N10
Classification msc 46C05
Classification msc 41A65
Classification msc 41A52
Classification msc 41A50