Linear and multilinear algebra; matrix theory Definition
Inner product space
Builds on Inner product
Definition
Let be or . An inner product space over is a vector space over together with an inner product on .
Remarks
A finite-dimensional real inner product space is a Euclidean space. Each inner product space has the induced norm . An inner product space that is complete for this norm is a Hilbert space. Each finite-dimensional inner product space is complete.
Reading path
To understand this page from the ground up, read these 3 pages in order. The foundations come first.
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- Vector space definition
- Inner product definition
- Inner product space definition
Where this leads
69 pages build on this one.
- Parallelogram law
- Parseval's identity
- Riesz representation theorem in finite dimensions
- Polarisation identity
- Spectral theorem
- Pythagorean theorem in an inner product space
- Least squares
- Eigenvalues of a self-adjoint operator are real
- Matrix of the adjoint is the conjugate transpose
- Spectral theorem for Hermitian matrices
- Normal operator
- Eigenspaces of a self-adjoint operator are orthogonal
- Norm induced by an inner product
- Self-adjoint operator
- Orthogonal projection is the closest point
- Principal axis theorem
- A linear isometry preserves the inner product
- Bessel's inequality
- Cauchy–Schwarz inequality
- Adjoint of a linear map
- 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
- Principal curvature
- Shape operator
- Response surface methodology
- Nonlinear regression
- Spectral theorem for compact self-adjoint operators
- Courant–Fischer characterization of characteristic values
- Generalized minimal residual method
- Minimum total potential energy principle
- Gauss–Newton algorithm
- Hilbert space adjoint
- Heisenberg inequality for the Fourier transform
- Lidskii's trace formula for integral operators
- Spectrum of the Laplace–Beltrami operator
- Hilbert projection theorem
- Ridge regression
- L-curve
- Biconjugate gradient stabilized method
- Galerkin method
- Bernstein's theorem on absolutely convergent Fourier series
- Operator monotone function
- Least squares approximation
- Empirical risk minimization
- Isotonic regression
- Mercer's theorem
- Positive definiteness of the energy
- Principal component analysis
- Ladyzhenskaya–Babuška–Brezzi condition
- Levenberg–Marquardt algorithm
- Schur product theorem
- Riesz representation theorem
- Box–Jenkins method
- Singular values of a compact operator
- Lasso
- Numerical methods for linear least squares
- Schmidt's singular value decomposition of a kernel
- Nesbitt's inequality
Further reading
About this page
| Tag | 01EK923SKP |
|---|---|
| URI | m2://01EK923SKP |
| Kind | definition |
| Writing | W1 stated |
| MSC | 15A63, 46C05 |
| Defines | Inner product space |
| Uses | Inner product |
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What this page needs for W2
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