Let 
be a Hilbert space and 
a closed subspace of 
. Corresponding to any vector 
, there is a unique vector 
such that
for all 
.
Furthermore, a necessary and sufficient condition that 
be the unique minimizing vector is that 
be orthogonal to 
(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
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References
Luenberger, D. G. Optimization by Vector Space Methods. New York: Wiley, 1997.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Projection Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ProjectionTheorem.html