Hilbert projection theorem

In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every point in a Hilbert space and every nonempty closed convex , there exists a unique point for which is minimized over .

This is, in particular, true for any closed subspace of . In that case, a necessary and sufficient condition for is that the vector be orthogonal to .

ProofEdit

  • Let us show the existence of y:

Let δ be the distance between x and C, (yn) a sequence in C such that the distance squared between x and yn is below or equal to δ2 + 1/n. Let n and m be two integers, then the following equalities are true:

 

and

 

We have therefore:

 

(Recall the formula for the median in a triangle - Median_(geometry)#Formulas_involving_the_medians'_lengths) By giving an upper bound to the first two terms of the equality and by noticing that the middle of yn and ym belong to C and has therefore a distance greater than or equal to δ from x, one gets :

 

The last inequality proves that (yn) is a Cauchy sequence. Since C is complete, the sequence is therefore convergent to a point y in C, whose distance from x is minimal.

  • Let us show the uniqueness of y :

Let y1 and y2 be two minimizers. Then:

 

Since   belongs to C, we have   and therefore

 

Hence  , which proves uniqueness.

  • Let us show the equivalent condition on y when C = M is a closed subspace.

The condition is sufficient: Let   such that   for all  .   which proves that   is a minimizer.

The condition is necessary: Let   be the minimizer. Let   and  .

 

is always non-negative. Therefore,  

QED

ReferencesEdit

  • Walter Rudin, Real and Complex Analysis. Third Edition, 1987.

See alsoEdit