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Projection Theorem


Let H be a Hilbert space and M a closed subspace of H. Corresponding to any vector x in H, there is a unique vector m_0 in M such that

 |x-m_0|<=|x-m|

for all m in M. Furthermore, a necessary and sufficient condition that m_0 in M be the unique minimizing vector is that x-m_0 be orthogonal to M (Luenberger 1997, p. 51).

This theorem can be viewed as a formalization of the result that the closest point on a plane to a point not on the plane can be found by dropping a perpendicular.


See also

Point-Plane Distance

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References

Luenberger, D. G. Optimization by Vector Space Methods. New York: Wiley, 1997.

Referenced on Wolfram|Alpha

Projection Theorem

Cite this as:

Weisstein, Eric W. "Projection Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProjectionTheorem.html

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