Math 2.0

Functional analysis Hilbert spaces Theorem

Bessel's inequality

Builds on Orthonormal familyNorm induced by an inner product

Statement

Let VV be an inner product space over ℝ\R or ℂ\C, with the induced norm ∥⋅∥\norm{\cdot}. Let e1,…,eke_1, \dots, e_k be an orthonormal family in VV. Then, for each x∈Vx \in V, ∑i=1k∣⟨x,ei⟩∣2≤∥x∥2.\sum_{i=1}^{k} \abs{\langle x, e_i \rangle}^2 \le \norm{x}^2.

Remarks

The proof applies the Pythagorean theorem to x=∑i⟨x,ei⟩ei+rx = \sum_i \langle x, e_i \rangle e_i + r, where rr is orthogonal to each eie_i. Equality holds if and only if xx lies in the span of e1,…,eke_1, \dots, e_k. The inequality also holds for infinite orthonormal families, with a series on the left.

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Vector space, definition, no evidence levelVector spacedefinitionInner product, definition, no evidence levelInner productdefinitionNorm, definition, no evidence levelNormdefinitionInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionNorm induced by an inner product, definition, no evidence levelNorm induced by an in…definitionOrthonormal family, definition, no evidence levelOrthonormal familydefinitionBessel's inequality, statement, no evidence levelBessel's inequalitystatement
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. Norm definition
  4. Inner product space definition
  5. Orthogonal vectors definition
  6. Norm induced by an inner product definition
  7. Orthonormal family definition
  8. Bessel's inequality statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Bessel's_inequality, Statement of the inequality
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Tag0RBT5PB74K
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Kindstatement, theorem
WritingW1 stated
MSC46C05
StatusThis copy of the site has no build data. Run m2 status 0RBT5PB74K.
UsesOrthonormal family, Norm induced by an inner product
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