Linked from

The 31 pages that link to Inner product, each with the reason it gives.

• Fourier transformRelated: The transform’s integral compares a function with oscillatory basis functions.

• Fourier seriesRelated: Integrals of products provide the projections used to calculate Fourier coefficients.

• Euclidean distanceRelated: The standard inner product determines the Euclidean norm used for distance.

• Cauchy–Schwarz inequalityNarrower topic: The inequality bounds the inner product relative to the lengths it defines.

• Metric tensorBroader topic: At each point, the metric is an inner product on the tangent space.

• Born ruleRelated: Measurement amplitudes are inner products between states and outcome vectors.

• OrthogonalityRelated: A pair is orthogonal precisely when this function evaluates to zero.

• Riemannian metricNarrower topic: A Riemannian metric is a smoothly varying family of inner products.

• Orthonormal basisNarrower topic: It defines both vector norms and the orthogonality required of the family.

• Inner product spaceBroader topic: This pairing supplies the lengths and angles associated with the space.

• Orthogonal matrixRelated: Preserving inner products is equivalent to preserving lengths and angles.

• Gram matrixNarrower topic: Each Gram-matrix entry is an inner product of two vectors.

• Riesz representation theoremRelated: The representing vector acts through this pairing to produce the functional's values.

• Fourier coefficientRelated: For orthogonal Fourier bases, an inner product extracts a component’s coefficient.

• Linear functionalCompared with: Fixing one argument can produce a functional, but an inner product itself takes two inputs.

• Bilinear formBroader topic: An inner product is a specialized bilinear form with positivity and symmetry.

• Slater determinantRelated: Orbital overlaps determine the normalization and overlaps of Slater determinants.

• Adjoint operatorNarrower topic: The adjoint is defined by moving an operator from one slot of this pairing to the other.

• Parallelogram lawRelated: Expanding the squared norms with an inner product makes the identity immediate.

• Positive-definite matrixRelated: The expression xᵀAx acts as a weighted squared length when A is positive definite.

• Conjugate gradient methodNarrower topic: Matrix-weighted inner products give precise meaning to conjugacy between search directions.

• No-cloning theoremBroader topic: Preservation of inner products makes distinct nonorthogonal states impossible to clone perfectly.

• Parseval's theoremNarrower topic: The norm side of Parseval’s theorem is determined by an inner product.

• Magnitude (mathematics)Related: An inner product induces a norm through the square root of a vector’s self-product.

• Euclidean vectorRelated: It defines the lengths and angles that make a vector space Euclidean.

• Fundamental theorem of linear algebraRelated: It provides the notion of orthogonality used by the theorem.

• Wigner's theoremRelated: Its magnitude determines the transition probabilities that the theorem requires symmetries to preserve.

• Titu's lemmaNarrower topic: Cauchy–Schwarz, and therefore Titu's lemma, bounds inner products.

• Bra–ket notationRelated: The expression ⟨φ|ψ⟩ represents an inner product.

• Hellinger–Toeplitz theoremNarrower topic: Symmetry is expressed through inner products of operator outputs and test vectors.

• No-deleting theoremBroader topic: The theorem follows by comparing input and output state overlaps.