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

• A conjugate point at which there are many geodesics Counterexample
• Bounded C1 domains and their outward normals 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
• Real and complex inner-product spaces and their induced length Definition
• Reduced crystallographic Euclidean root system Definition
• Smooth bundle metrics Definition
• The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V 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 norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product Definition
• The Rayleigh quotient of a nonzero vector for a self-adjoint endomorphism Definition
• The standard inner product on cf(G) Definition
• 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
• 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
• FALSE: ⟨ x,y⟩=∑ᵢxᵢyᵢ makes F₂ⁿ an inner product space False statement
• FALSE: Every complex inner product satisfies ⟨ u,v⟩=⟨ v,u⟩ False statement
• If v₁,…,vₘ∈ℝⁿ satisfy ⟨ vᵢ,vⱼ⟩=t≥0 for i≠ j and ⟨ vᵢ,vᵢ⟩>t, they are linearly independent Lemma
• Inner products separate vectors, and the induced norm is homogeneous: ‖λ v‖=|λ|‖ v‖ Lemma
• The normalized Hermitian form on a finite function space Lemma
• The quaternion double cover generates the third homotopy group of SO(3) Lemma
• Two dimensional numerical range is convex Lemma
• Orthogonal projection is linear, and an orthonormal basis (eᵢ) of W gives P_Wv=∑ᵢ⟨ v,eᵢ⟩ eᵢ Proposition
• Pythagoras, the parallelogram identity, and the real and complex polarisation identities Proposition
• The standard formulas ⟨ x,y⟩=∑_k<nxₖ yₖ on ℝⁿ and ∑_k<nxₖyₖ on ℂⁿ are inner products Proposition
• Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors Theorem
• Every finite orthogonal list of nonzero vectors is linearly independent Theorem
• Hilbert spaces are reflexive by Riesz representation Theorem
• Hᵏ is a Hilbert space under the derivative-sum inner product Theorem
• Numerical radius is an equivalent operator norm Theorem
• Partial isometry characterizations Theorem
• The Gram formula gives a well-defined positive-definite inner product on exterior powers, and ‖v₁∧⋯∧ vₖ‖² is the Gram determinant Theorem

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)