Math 2.0

Functional analysis Proposition

Orthogonal complement of the orthogonal complement

Builds on Orthogonal complementDimension of the orthogonal complement

Statement

Let VV be a finite-dimensional inner product space over ℝ\R or ℂ\C, and let WW be a subspace of VV. Then (W⊥)⊥=W,(W^\perp)^\perp = W, where W⊥W^\perp is the orthogonal complement of WW.

Remarks

Each vector of the subspace is orthogonal to each vector of its orthogonal complement, so W⊆(W⊥)⊥W \subseteq (W^\perp)^\perp. Both have dimension dim⁡V−dim⁡W⊥=dim⁡W\dim V - \dim W^\perp = \dim W. In an infinite-dimensional Hilbert space, (W⊥)⊥(W^\perp)^\perp is the closure of WW.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionInner product, definition, no evidence levelInner productdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionOrthogonal complement, definition, no evidence levelOrthogonal complementdefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionDimension, definition, no evidence levelDimensiondefinitionDimension of the orthogonal complement, statement, no evidence levelDimension of the orth…statementOrthogonal complement of the orthogonal complement, statement, no evidence levelOrthogonal complement…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.
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  1. Vector space definition
  2. Inner product definition
  3. Linear combination definition
  4. Linear subspace definition
  5. Orthogonal vectors definition
  6. Linear independence definition
  7. Linear span definition
  8. Orthogonal complement definition
  9. Spanning set definition
  10. Basis definition
  11. Finite-dimensional vector space definition
  12. Dimension definition
  13. Dimension of the orthogonal complement statement
  14. Orthogonal complement of the orthogonal complement statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Orthogonal_complement, Finite dimensions
About this page
Tag0PXN9G4MX1
URIm2://0PXN9G4MX1
Kindstatement, proposition
WritingW0 stub
MSC46C05
StatusThis copy of the site has no build data. Run m2 status 0PXN9G4MX1.
UsesOrthogonal complement, Dimension of the orthogonal complement

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