Math 2.0

Functional analysis Theorem

Orthogonal projection is the closest point

Builds on Orthogonal projectionNorm induced by an inner product

Statement

Let VV be an inner product space over ℝ\R or ℂ\C, let WW be a finite-dimensional subspace of VV, and let v∈Vv \in V. Let PWP_W be the orthogonal projection onto WW. Then, for all w∈Ww \in W, ∥v−PW(v)∥≤∥v−w∥,\norm{v - P_W(v)} \le \norm{v - w}, with equality only for w=PW(v)w = P_W(v).

Remarks

The proof: v−w=(v−PW(v))+(PW(v)−w)v - w = (v - P_W(v)) + (P_W(v) - w) is a sum of orthogonal vectors, so the Pythagorean theorem applies. The number ∥v−PW(v)∥\norm{v - P_W(v)} is the distance from vv to WW. The Hilbert projection theorem gives the same result for closed convex sets in Hilbert spaces. This is the basis of the method of least squares.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionInner product, definition, no evidence levelInner productdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionNorm, definition, no evidence levelNormdefinitionInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionSum of subspaces, definition, no evidence levelSum of subspacesdefinitionNorm induced by an inner product, definition, no evidence levelNorm induced by an in…definitionOrthogonal complement, definition, no evidence levelOrthogonal complementdefinitionDirect sum of subspaces, definition, no evidence levelDirect sum of subspac…definitionA space is the direct sum of a subspace and its orthogonal complement, statement, no evidence levelA space is the direct…statementProjection, definition, no evidence levelProjectiondefinitionOrthogonal projection, definition, no evidence levelOrthogonal projectiondefinitionOrthogonal projection is the closest point, statement, no evidence levelOrthogonal projection…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Vector space definition
  2. Inner product definition
  3. Linear map definition
  4. Linear subspace definition
  5. Norm definition
  6. Inner product space definition
  7. Orthogonal vectors definition
  8. Endomorphism definition
  9. Sum of subspaces definition
  10. Norm induced by an inner product definition
  11. Orthogonal complement definition
  12. Direct sum of subspaces definition
  13. A space is the direct sum of a subspace and its orthogonal complement statement
  14. Projection definition
  15. Orthogonal projection definition
  16. Orthogonal projection is the closest point statement

Where this leads

28 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert_projection_theorem, Finite dimensional case
About this page
Tag0MCDDSCB9Z
URIm2://0MCDDSCB9Z
Kindstatement, theorem
WritingW0 stub
MSC46C05, 65F20
StatusThis copy of the site has no build data. Run m2 status 0MCDDSCB9Z.
UsesOrthogonal projection, Norm induced by an inner product

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