Functional analysis Definition
Norm induced by an inner product
Builds on Inner product spaceNorm
Definition
Let be an inner product space over or , with inner product . The norm induced by the inner product is the map
Remarks
It is a norm; the triangle inequality follows from the Cauchy–Schwarz inequality. For the dot product on , it is the Euclidean length . It satisfies the parallelogram law, and the polarisation identity recovers the inner product from it.
Reading path
To understand this page from the ground up, read these 5 pages in order. The foundations come first.
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- Vector space definition
- Inner product definition
- Norm definition
- Inner product space definition
- Norm induced by an inner product definition
Where this leads
46 pages build on this one.
- Parallelogram law
- Parseval's identity
- Polarisation identity
- Pythagorean theorem in an inner product space
- Least squares
- Orthogonal projection is the closest point
- A linear isometry preserves the inner product
- Bessel's inequality
- Cauchy–Schwarz inequality
- Non-negative least squares
- Clarkson's inequalities
- Paley–Zygmund inequality
- Conjugate residual method
- Local regression
- Hilbert–Schmidt expansion theorem
- Minimal residual method
- Importance sampling
- Céa's lemma
- Bilinear series of a symmetric kernel
- Ordinary least squares
- Smoothing spline
- Lax–Milgram theorem
- Response surface methodology
- Nonlinear regression
- Generalized minimal residual method
- Minimum total potential energy principle
- Gauss–Newton algorithm
- Hilbert space adjoint
- Heisenberg inequality for the Fourier transform
- Hilbert projection theorem
- Ridge regression
- L-curve
- Biconjugate gradient stabilized method
- Galerkin method
- Bernstein's theorem on absolutely convergent Fourier series
- Least squares approximation
- Empirical risk minimization
- Isotonic regression
- Positive definiteness of the energy
- Ladyzhenskaya–Babuška–Brezzi condition
- Levenberg–Marquardt algorithm
- Riesz representation theorem
- Box–Jenkins method
- Lasso
- Numerical methods for linear least squares
- Nesbitt's inequality
Further reading
- Web https://en.wikipedia.org/wiki/Inner_product_space, Norm properties
About this page
| Tag | 0KG4D8AP6G |
|---|---|
| URI | m2://0KG4D8AP6G |
| Kind | definition |
| Writing | W0 stub |
| MSC | 46C05 |
| Defines | Norm induced by an inner product |
| Uses | Inner product space, Norm |
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