Real and complex inner product spaces, with the inner product linear in the first argument

Depends on

• Sesquilinear and Hermitian forms over a field with an involution, using the convention linear in the first variable
• Real and imaginary parts, complex conjugation, and modulus

Used by

• An inverse-error chain obeys Minkowski in L²; root-sum-square is equivalent to aggregate cross-term cancellation Corollary
• A conjugate point at which there are many geodesics Counterexample
• Affine root hyperplanes, coroot translations, alcoves, and the affine reflection group Definition
• Bounded C1 domains and their outward normals Definition
• Exact correction of a protected subsystem Definition
• Exact recovery of an encoded quantum code Definition
• Filtered γ₂ factorization norm for query mismatch masks 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
• Partial trace over the first tensor factor Definition
• Positive semidefinite kernels and reproducing kernel Hilbert spaces Definition
• Real and complex inner-product spaces and their induced length Definition
• Real spin-factor product Definition
• Reduced crystallographic Euclidean root system Definition
• Score and Fisher information Definition
• Smooth bundle metrics Definition
• The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V Definition
• The B-inner product, B-norm, and dual residual norm Definition
• The counting inner product on ℂ^ℤ/Nℤ Definition
• The Gram inner product on ΛᵏV Definition
• The Gram matrix G(v₀,…,vᵣ₋₁)=(⟨ vᵢ,vⱼ⟩)_i,j<r and Gram determinant, with empty value 1 Definition
• The neural tangent kernel of a differentiable parameterized function Definition
• The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product Definition
• The polynomial moment pairing Definition
• The Rayleigh quotient of a nonzero vector for a self-adjoint endomorphism Definition
• The standard inner product on cf(G) Definition
• A degenerate code with a non-diagonal Knill–Laflamme matrix Example
• Chebyshev extremals and the exact disk Fekete polynomial Example
• Evaluation at four distinct real points gives an inner product on polynomials of degree at most three Example
• Hilbert spaces are uniformly convex Example
• Norm-attaining functionals on a Hilbert space Example
• Orthogonal error spaces and a recovery branch Example
• Positive coordinate weights define an inner product and change lengths and projections Example
• The Frobenius inner product on a real or complex matrix space Example
• The hexagonal A₂ cell: Euclidean cell metric versus graph distance Example
• Underestimated noise can put the Morozov target below every residual Example
• FALSE: ⟨ x,y⟩=∑ᵢxᵢyᵢ makes F₂ⁿ an inner product space False statement
• FALSE: Every complex inner product satisfies ⟨ u,v⟩=⟨ v,u⟩ False statement
• A compact Hilbert-space Gram operator is positive and has the forward kernel Lemma
• A finite reversible transition operator is self-adjoint on weighted L2 Lemma

…and 57 more results.

Dependency tree · two levels

7 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)