The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product

Depends on

• Real and complex inner product spaces, with the inner product linear in the first argument
• Existence and uniqueness of $n$-th roots: a unique $a^{1/n} \ge 0$ with $(a^{1/n})^n = a$

Used by

• The induced length is a norm Corollary
• The inner-product norm is definite, homogeneous, and satisfies the triangle inequality Corollary
• A conjugate point at which there are many geodesics Counterexample
• Householder reflectors in real or complex inner-product spaces Definition
• Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces Definition
• Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases Definition
• Rayleigh-quotient iteration for Hermitian matrices Definition
• Real and complex inner-product spaces and their induced length Definition
• Real Givens rotations and complex Givens transformations Definition
• The Arnoldi process for building an orthonormal basis of a Krylov subspace Definition
• The Frobenius norm ‖ A‖_F=(∑_i,j|aᵢⱼ|²)^1/2 on real or complex matrices Definition
• The normwise condition number of a simple eigenvalue Definition
• The operator norm is zero on the zero domain and otherwise is max_||v||=1 ||Tv|| Definition
• The residual r=Ax-μ x and the normwise backward error of an approximate eigenpair Definition
• The Tikhonov regularised least-squares objective ‖Ax-b‖₂²+λ‖x‖₂² for λ>0 Definition
• Hilbert spaces are uniformly convex Example
• Norm-attaining functionals on a Hilbert space Example
• Positive coordinate weights define an inner product and change lengths and projections Example
• Inner products separate vectors, and the induced norm is homogeneous: ‖λ v‖=|λ|‖ v‖ Lemma
• The quaternion double cover generates the third homotopy group of SO(3) Lemma
• Pythagoras, the parallelogram identity, and the real and complex polarisation identities Proposition
• Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors Theorem
• Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs Theorem
• Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans Theorem
• Hilbert spaces are reflexive by Riesz representation Theorem
• Jordan–von Neumann: a norm is induced by an inner product exactly when it satisfies the parallelogram law Theorem
• The parallelogram law Theorem

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Sheldon Axler, Linear Algebra Done Right, 4th ed., §6A (standard reference, not scraped)
• Sergei Treil, Linear Algebra Done Wrong, Ch. 5, §5.1 (standard reference, not scraped)