The inner-product norm is definite, homogeneous, and satisfies the triangle inequality

Depends on

• The norm $\lVert v\rVert=\sqrt{\langle v,v\rangle}$ induced by a real or complex inner product
• Inner products separate vectors, and the induced norm is homogeneous: $\lVert\lambda v\rVert=|\lambda|\lVert v\rVert$
• Cauchy–Schwarz: $|\langle u,v\rangle|\leq\lVert u\rVert\lVert v\rVert$, with equality exactly for linearly dependent vectors
• Real and imaginary parts, complex conjugation, and modulus

Used by

• Qubitized Hamiltonian simulator Algorithm
• Measurable and decomposable operator fields Definition
• Projective braid gate and operator metric Definition
• The Frobenius norm ‖ A‖_F=(∑_i,j|aᵢⱼ|²)^1/2 on real or complex matrices Definition
• A Three-Factor Solovay–Kitaev Error Budget Example
• Cut locus on a flat rectangular torus from the Dirichlet cell Example
• Hilbert spaces are uniformly convex Example
• Norm-attaining functionals on a Hilbert space Example
• The claim that the ball and the polydisc are biholomorphic False statement
• A measurable direct integral of unitary representations is strongly continuous Lemma
• A proper matrix-product cover from the mixed (2,1) norm Lemma
• Bounded bowl-shaped losses have Gaussian-centered Lipschitz minorants Lemma
• Finite tie-broken projection searches are Borel measurable Lemma
• Huber covariance modulus lower bound for unrestricted matrix-valued actions Lemma
• LMNN hinge subgradient formula and a uniform data-dependent bound Lemma
• Normal relative property (T) controls the distance to the invariant subspace Lemma
• Normalization is stable under a small flagged-vector error Lemma
• Operator error for the finite sine-prefix rotation table Lemma
• Operator perturbation bound for QSVT Lemma
• Robust one-step OAA including ancilla leakage Lemma
• The quaternion double cover generates the third homotopy group of SO(3) Lemma
• The real positive-semidefinite cone is closed in the Frobenius metric Lemma
• Uniform deviation for squared-projection slabs Lemma
• Amplitude Table Precision and Compilation Theorem
• Direct integrals of measurable Hilbert fields are Hilbert spaces Theorem
• Element-distinctness gate cost in the QRAM model Theorem
• Finite adiabatic path simulation from term access Theorem
• Finite reversible-walk finding cost Theorem
• Finite universal-gate compilation cost for the truncated QFT Theorem
• Finiteness criterion: W is finite exactly when the Coxeter form is positive definite Theorem
• Fixed-Arity Unitary Error Budget Theorem
• HHL sparse-query and state-preparation cost Theorem
• Hybrid bound for imperfect QSP phase rotations Theorem
• LCU projected-block correctness Theorem
• Matrix-function transformation error and resource bound Theorem
• Measurable essentially bounded operator fields act decomposably Theorem
• Operator-norm error of QFT truncation Theorem
• QC24 Discrete Alphabet Approximate Preparation Lower Bound Theorem
• QSVT robustness, approximation error, and resources Theorem
• Qubitized simulation correctness Theorem

…and 5 more results.

Dependency tree · two levels

11 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)