Math 2.0

Functional analysis Hilbert spaces Theorem

Parseval's identity

Builds on Orthonormal basisNorm induced by an inner product

Statement

Let VV be a finite-dimensional inner product space over ℝ\R or ℂ\C, and let (e1,…,en)(e_1, \dots, e_n) be an orthonormal basis of VV. Then, for all x,y∈Vx, y \in V, ⟨x,y⟩=∑i=1n⟨x,ei⟩⟨y,ei⟩‾and∥x∥2=∑i=1n∣⟨x,ei⟩∣2.\langle x, y \rangle = \sum_{i=1}^{n} \langle x, e_i \rangle \overline{\langle y, e_i \rangle} \qquad \text{and} \qquad \norm{x}^2 = \sum_{i=1}^{n} \abs{\langle x, e_i \rangle}^2.

Remarks

It is the case of equality in Bessel's inequality. In coordinates of an orthonormal basis, the inner product becomes the standard inner product of 𝕂n\mathbb{K}^n. For Fourier series, the same identity holds with infinite sums, for square integrable functions.

Reading path

To understand this page from the ground up, read these 15 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 subspacedefinitionNorm, definition, no evidence levelNormdefinitionInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionOrthonormal family, definition, no evidence levelOrthonormal familydefinitionNorm induced by an inner product, definition, no evidence levelNorm induced by an in…definitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionOrthonormal basis, definition, no evidence levelOrthonormal basisdefinitionParseval's identity, statement, no evidence levelParseval's identitystatement
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 combination definition
  4. Linear subspace definition
  5. Norm definition
  6. Inner product space definition
  7. Orthogonal vectors definition
  8. Linear independence definition
  9. Linear span definition
  10. Orthonormal family definition
  11. Norm induced by an inner product definition
  12. Spanning set definition
  13. Basis definition
  14. Orthonormal basis definition
  15. Parseval's identity statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Parseval's_identity, lead section
About this page
Tag047RFCNVSK
URIm2://047RFCNVSK
Kindstatement, theorem
WritingW1 stated
MSC46C05
StatusThis copy of the site has no build data. Run m2 status 047RFCNVSK.
UsesOrthonormal basis, Norm induced by an inner product
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