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In mathematics, the '''Hilbert projection theorem''' is a famous result of [[convex analysis]] that says that for every point <math>x</math> in a [[Hilbert space]] <math>H</math> and every closed convex <math>C \subset H</math>, there exists a unique point <math>y \in C</math> for which <math>\lVert x - y \rVert</math> is minimized over <math>C</math>. | |||
This is, in particular, true for any closed subspace <math>M</math> of <math>H</math>. In that case, a necessary and sufficient condition for <math>y</math> is that the vector <math> x-y</math> be orthogonal to <math>M</math>. | |||
==Proof== | |||
:* ''Let us show the existence of ''y'':'' | |||
Let δ be the distance between ''x'' and ''C'', (''y''<sub>''n''</sub>) a sequence in ''C'' such that the distance squared between ''x'' and ''y''<sub>''n''</sub> is below or equal to δ<sup>2</sup> + 1/''n''. Let ''n'' and ''m'' be two integers, then the following equalities are true: | |||
: <math>\| y_n - y_m \|^2 = \|y_n -x\|^2 + \|y_m -x\|^2 - 2 \langle y_n - x \, , \, y_m - x\rangle</math> | |||
and | |||
: <math>4 \| \frac{y_n + y_m}2 -x \|^2 = \|y_n -x\|^2 + \|y_m -x\|^2 + 2 \langle y_n - x \, , \, y_m - x\rangle</math> | |||
We have therefore: | |||
: <math>\| y_n - y_m \|^2 = 2\|y_n -x\|^2 + 2\|y_m -x\|^2 - 4\| \frac{y_n + y_m}2 -x \|^2</math> | |||
By giving an upper bound to the first two terms of the equality and by noticing that the middle of ''y''<sub>''n''</sub> and ''y''<sub>''m''</sub> belong to ''C'' and has therefore a distance greater than or equal to ''δ'' from ''x'', one gets : | |||
: <math>\| y_n - y_m \|^2 \; \le \; 2\left(\delta^2 + \frac 1n\right) + 2\left(\delta^2 + \frac 1m\right) - 4\delta^2=2\left( \frac 1n + \frac 1m\right)</math> | |||
The last inequality proves that (''y''<sub>''n''</sub>) 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 ''y''<sub>1</sub> and ''y''<sub>2</sub> be two minimizer. Then: | |||
: <math>\| y_2 - y_1 \|^2 = 2\|y_1 -x\|^2 + 2\|y_2 -x\|^2 - 4\| \frac{y_1 + y_2}2 -x \|^2</math> | |||
Since <math>\frac{y_1 + y_2}2</math> belongs to ''C'', we have <math>\| \frac{y_1 + y_2}2 -x \|^2\geq \delta^2</math> and therefore | |||
: <math>\| y_2 - y_1 \|^2 \leq 2\delta^2 + 2\delta^2 - 4\delta^2=0 \, </math> | |||
Hence <math>y_1=y_2</math>, which proves unicity. | |||
:* ''Let us show the equivalent condition on'' ''y'' ''when'' ''C'' = ''M'' ''is a closed subspace.'' | |||
The condition is sufficient: | |||
Let <math>z\in M</math> such that <math>\langle z-x, a \rangle=0</math> for all <math>a\in M</math>. | |||
<math>\|x-a\|^2=\|z-x\|^2+\|a-z\|^2+2\langle z-x, a-z \rangle=\|z-x\|^2+\|a-z\|^2</math> which proves that <math>z</math> is a minimizer. | |||
The condition is necessary: | |||
Let <math>y\in M</math> be the minimizer. Let <math>a\in M</math> and <math>t\in\mathbb R</math>. | |||
: <math>\|(y+t a)-x\|^2-\|y-x\|^2=2t\langle y-x,a\rangle+t^2 \|a\|^2=2t\langle y-x,a\rangle+O(t^2)</math> | |||
is always non-negative. Therefore, <math>\langle y-x,a\rangle=0.</math> | |||
QED | |||
==References== | |||
* [[Walter Rudin]], ''Real and Complex Analysis. Third Edition'', '''1987'''. | |||
==See also== | |||
*[[Orthogonality principle]] | |||
{{DEFAULTSORT:Hilbert Projection Theorem}} | |||
[[Category:Convex analysis]] | |||
[[Category:Theorems in functional analysis]] |
Latest revision as of 19:58, 16 March 2013
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 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 .
Proof
- 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:
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 minimizer. Then:
Since belongs to C, we have and therefore
- 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
References
- Walter Rudin, Real and Complex Analysis. Third Edition, 1987.