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Real and complex inner product spaces, with the inner product linear in the first argument
Definition
Let be either or , with conjugation equal to the identity on and with complex conjugation as in Real and imaginary parts, complex conjugation, and modulus. An inner product on an -vector space is a function such that for all and :
- ;
- ;
- is real and nonnegative, and if and only if .
The first two clauses imply conjugate-linearity in the second argument: . This is the linear-first convention of Sesquilinear and Hermitian forms over a field with an involution, using the convention linear in the first variable. A vector space equipped with an inner product is an inner product space.
Depends on
Used by
- 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
- The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V 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
- Evaluation at four distinct real points gives an inner product on polynomials of degree at most three 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: Every complex inner product satisfies ⟨ u,v⟩=⟨ v,u⟩ False statement
- Inner products separate vectors, and the induced norm is homogeneous: ‖λ v‖=|λ|‖ v‖ 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<nxₖ overlineyₖ 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
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 28 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)