Math 2.0

Functional analysis Proof

Proof that the orthogonal complement of the orthogonal complement is the subspace

Builds on Inner productOrthogonal complementDimension of the orthogonal complementA subspace of full dimension is the whole spaceDimension of a subspace

Proof

We show one inclusion, and then compare dimensions.

  1. Let VV be a finite-dimensional inner product space over ℝ\R or ℂ\C, with inner product ⟨⋅,⋅⟩\ip{\cdot}{\cdot}. Let WW be a subspace of VV and W⊥W^\perp its orthogonal complement. Set n=dim⁡Vn = \dim V.
  2. Let w∈Ww \in W and x∈W⊥x \in W^\perp. By conjugate symmetry, ⟨w,x⟩=⟨x,w⟩‾=0‾=0\ip{w}{x} = \overline{\ip{x}{w}} = \overline{0} = 0.
  3. So ww is orthogonal to every vector of W⊥W^\perp, that is, w∈(W⊥)⊥w \in (W^\perp)^\perp. Hence W⊆(W⊥)⊥W \subseteq (W^\perp)^\perp.
  4. The set W⊥W^\perp contains 00 and is closed under linear combinations, by linearity in the first argument. So it is a subspace of VV. By the dimension formula, dim⁡W⊥=n−dim⁡W\dim W^\perp = n - \dim W.
  5. The same formula for the subspace W⊥W^\perp gives dim⁡(W⊥)⊥=n−dim⁡W⊥=dim⁡W\dim (W^\perp)^\perp = n - \dim W^\perp = \dim W.
  6. By step 3, WW is a subspace of the finite-dimensional space (W⊥)⊥(W^\perp)^\perp, and by step 5 it has the same dimension.
  7. A subspace of full dimension is the whole space. Hence W=(W⊥)⊥W = (W^\perp)^\perp.

Idea

Each vector of WW is orthogonal to each vector of W⊥W^\perp, so WW lies inside (W⊥)⊥(W^\perp)^\perp. Taking the complement twice brings the dimension back to dim⁡W\dim W. A subspace inside another one of the same finite dimension is equal to it.

Reading path

To understand this page from the ground up, read these 78 pages in order. The foundations come first. The path takes the proofs Proof of the Steinitz exchange lemma by induction, Proof that a subspace of full dimension is the whole space, Proof of the existence of an orthonormal basis by the Gram–Schmidt process, Proof of Grassmann's formula by extending a basis of the intersection, Proof that the Gram–Schmidt vectors are orthogonal, Proof that an orthonormal family is linearly independent, Proof of the formula for the dimension of the orthogonal complement, Proof that every linearly independent set extends to a basis, by Zorn's lemma, Proof of the formula for coordinates in an orthonormal basis, Proof of Zorn's lemma, Proof that n independent or n spanning vectors in an n-dimensional space form a basis, Proof that a space is the direct sum of a finite-dimensional subspace and its orthogonal complement, Proof that a subspace of a finite-dimensional space has smaller or equal dimension, Proof that an intersection of subspaces is a subspace.

  1. Element definition
  2. Proof of Zorn's lemma proof
  3. Proposition concept
  4. Vector space definition
  5. Set definition
  6. Disjunction definition
  7. Implication definition
  8. Predicate definition
  9. Inner product definition
  10. Linear combination definition
  11. Linear subspace definition
  12. Norm definition
  13. Family of sets definition
  14. Ordered pair definition
  15. Union definition
  16. Existential quantifier definition
  17. Universal quantifier definition
  18. Inner product space definition
  19. Orthogonal vectors definition
  20. Linear independence definition
  21. Linear span definition
  22. Proof that an intersection of subspaces is a subspace proof
  23. Sum of subspaces definition
  24. Cartesian product definition
  25. Subset definition
  26. Union of a family of sets definition
  27. Uniqueness quantifier definition
  28. Norm induced by an inner product definition
  29. Orthogonal complement definition
  30. Orthonormal family definition
  31. Intersection of subspaces statement
  32. Spanning set definition
  33. Direct sum of subspaces definition
  34. Binary relation definition
  35. Proof that an orthonormal family is linearly independent proof
  36. Gram–Schmidt process concept
  37. Proof of the Steinitz exchange lemma by induction proof
  38. Basis definition
  39. Finite-dimensional vector space definition
  40. Antisymmetric relation definition
  41. Reflexive relation definition
  42. Transitive relation definition
  43. Function definition
  44. Dimension theorem for vector spaces statement
  45. Dimension definition
  46. An orthonormal family is linearly independent statement
  47. Orthonormal basis definition
  48. Proof that the Gram–Schmidt vectors are orthogonal proof
  49. Steinitz exchange lemma statement
  50. Preorder definition
  51. Choice function definition
  52. Proof of the formula for coordinates in an orthonormal basis proof
  53. Gram–Schmidt vectors are orthogonal statement
  54. Proof that a subspace of a finite-dimensional space has smaller or equal dimension proof
  55. Proof that n independent or n spanning vectors in an n-dimensional space form a basis proof
  56. Partial order definition
  57. Axiom of choice statement
  58. Coordinates in an orthonormal basis statement
  59. Dimension of a subspace statement
  60. Linearly independent set of the right size is a basis statement
  61. Maximal element definition
  62. Total order definition
  63. Upper bound definition
  64. Proof that a subspace of full dimension is the whole space proof
  65. Chain definition
  66. A subspace of full dimension is the whole space statement
  67. Zorn's lemma statement
  68. Proof that every linearly independent set extends to a basis, by Zorn's lemma proof
  69. Every linearly independent set extends to a basis statement
  70. Proof of the existence of an orthonormal basis by the Gram–Schmidt process proof
  71. Proof of Grassmann's formula by extending a basis of the intersection proof
  72. Existence of an orthonormal basis statement
  73. Grassmann's formula statement
  74. Proof that a space is the direct sum of a finite-dimensional subspace and its orthogonal complement proof
  75. A space is the direct sum of a subspace and its orthogonal complement statement
  76. Proof of the formula for the dimension of the orthogonal complement proof
  77. Dimension of the orthogonal complement statement
  78. Proof that the orthogonal complement of the orthogonal complement is the subspace proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://linear.axler.net/LADR4e.pdf, Theorem 6.52
About this page
TagFZEF1XYTVB
URIm2://FZEF1XYTVB
Kindproof
WritingW0 stub
MSC46C05
ProvesOrthogonal complement of the orthogonal complement
UsesInner product, Orthogonal complement, Dimension of the orthogonal complement, A subspace of full dimension is the whole space, Dimension of a subspace
Changed4 November 2026

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