Math 2.0

Functional analysis Theorem

A space is the direct sum of a subspace and its orthogonal complement

Builds on Orthogonal complementDirect sum of subspaces

Statement

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

Remarks

For an orthonormal basis e1,…,eke_1, \dots, e_k of WW, each v∈Vv \in V is w+(v−w)w + (v - w) with w=∑i⟨v,ei⟩ei∈Ww = \sum_i \langle v, e_i \rangle e_i \in W and v−w∈W⊥v - w \in W^\perp. The statement holds for each closed subspace of a Hilbert space. It can fail for a subspace that is not closed in an infinite-dimensional space.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionInner product, definition, no evidence levelInner productdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionSum of subspaces, definition, no evidence levelSum of subspacesdefinitionOrthogonal 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…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 subspace definition
  4. Orthogonal vectors definition
  5. Sum of subspaces definition
  6. Orthogonal complement definition
  7. Direct sum of subspaces definition
  8. A space is the direct sum of a subspace and its orthogonal complement statement

Where this leads

35 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Orthogonal_complement, Properties
About this page
Tag0SP7P0G7KQ
URIm2://0SP7P0G7KQ
Kindstatement, theorem
WritingW0 stub
MSC46C05
StatusThis copy of the site has no build data. Run m2 status 0SP7P0G7KQ.
UsesOrthogonal complement, Direct sum of subspaces

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