scalar product

Declared product sense

Treat positive-definite inner products as the core sense and explicitly identify broader uses of the term.

1. Does scalar product here mean the Euclidean dot product, an arbitrary positive-definite inner product or a broader pairing? definition
2. Which reference or application specification establishes this meaning? provenance

Compatible admissible vectors

A product requires vectors in its declared space; function-space products may require integrability and equivalence-class conventions.

1. What vector space and scalar field contain both operands? definition
2. For function operands, what integrability requirements and identifications up to equality almost everywhere make the product well-defined and positive definite? boundary

Field-sensitive algebra

Real products are symmetric bilinear forms; complex products are conjugate symmetric with opposite linearity behaviour in their two arguments.

1. Which argument is linear, and where must scalar coefficients be conjugated? definition
2. What proof or defining construction establishes additivity, scalar compatibility and the required symmetry? provenance

Self-product test

An inner product has a real, strictly positive self-product for every nonzero vector; semidefinite forms may instead induce products on a quotient by their null space.

1. Can any nonzero admissible vector have zero, negative or nonreal self-product? boundary
2. If a null space exists, is the intended object a semidefinite form or an inner product on an explicitly defined quotient space? definition

Gram-matrix contract

For column coordinates and the complex convention linear in the second argument, the product is x*Gy, where G is Hermitian positive definite; real coordinates use xᵀGy.

1. Which ordered basis and Gram matrix define the coordinate formula, and is that basis orthonormal for this product? definition
2. Under a change of basis, how are coordinates and the Gram matrix transformed to preserve the scalar value? action

Evaluation rule and existence

Weighted sums and integrals require explicit weights, conjugation, measures and existence conditions.

1. What sum or integral computes the product, including its weights, measure and conjugated argument? definition
2. What guarantees convergence and positive definiteness on the declared operand space? boundary

Geometric interpretation

The induced norm is the square root of the self-product and orthogonality means a zero product; the usual cosine angle formula applies directly to nonzero real vectors.

1. What lengths and orthogonality relation does this product induce for the operands? measurement
2. If an angle is requested, are both vectors nonzero, and what angle convention is declared for a complex space? boundary

Projection preconditions

Projection onto a nonzero vector uses its positive self-product as a denominator; projection onto general subspaces requires an existence argument, with closed subspaces of Hilbert spaces providing a standard guarantee.

1. Which normalisation or projection formula matches the declared argument convention, and are its denominators nonzero? action
2. For a requested subspace projection, what establishes existence and uniqueness in the ambient space? boundary

Near-zero product

A small computed value supports approximate orthogonality only relative to operand norms, product scaling and evaluation error.

1. What precision, accumulation method and error estimate support the evaluated product? measurement
2. What scale-aware tolerance justifies treating the operands as approximately orthogonal? action

Weighting, units and normalisation

A raw product combines magnitude with alignment; weights and units come from the application, while normalisation changes the quantity being interpreted.

1. Why was this product or weighting selected, and what units does its output carry? provenance
2. Does the intended decision require a raw product, a normalised similarity or a projection coefficient? action

Check these first

Recalled without web access and unsourced; every item is a lead to verify.

• The registry supplies no sense: check whether the intended scope is the Euclidean dot product, general inner products, or a broader usage that includes indefinite forms.
• Mathematical and physics sources use different conventions for which argument of a complex inner product is linear.
• This description is recalled knowledge; no sources or standards were consulted.

1. Which of these check these first hold for the sense of scalar product this model covers, and on what evidence? provenance

Kinds and varieties

Recalled without web access and unsourced; every item is a lead to verify.

• Euclidean dot product
• Weighted inner product defined by a positive-definite matrix
• Complex Hermitian inner product
• Integral inner product on a space of functions

1. Which of these kinds and varieties hold for the sense of scalar product this model covers, and on what evidence? provenance

Real-world use

Recalled without web access and unsourced; every item is a lead to verify.

• Computing lengths, angles, orthogonality and projections in geometry.
• Expressing mechanical work as the dot product of a constant force and displacement.
• Constructing least-squares approximations and orthogonal decompositions.
• Comparing vector representations through cosine similarity.
• Calculating overlaps between quantum states using complex inner products.

1. Which of these real-world use hold for the sense of scalar product this model covers, and on what evidence? provenance

Typical measurements

Recalled without web access and unsourced; every item is a lead to verify.

• Euclidean dot product of vectors u and v - Between -||u|| ||v|| and +||u|| ||v||; no universal numerical range - Product of the units of the two vectors, when their components have compatible units
• Normalized real inner product for nonzero vectors - [-1, 1] - Dimensionless

1. Which of these typical measurements hold for the sense of scalar product this model covers, and on what evidence? provenance

Failure modes and hazards

Recalled without web access and unsourced; every item is a lead to verify.

• Using the coordinate formula sum(u_i v_i) in a nonorthonormal basis without the required Gram matrix.
• Omitting complex conjugation for complex vectors, thereby losing positive definiteness.
• Mixing conventions about which argument of a complex inner product is linear.
• Normalizing a zero vector, for which cosine similarity is undefined.
• Treating an indefinite form as a positive-definite inner product, leading to invalid conclusions about lengths or angles.

1. Which of these failure modes and hazards hold for the sense of scalar product this model covers, and on what evidence? provenance

Neighbouring kinds and how to tell them apart

Recalled without web access and unsourced; every item is a lead to verify.

• Scalar multiplication - Combines a scalar and a vector to produce a vector; a scalar product combines two vectors to produce a scalar.
• Dot product - Usually denotes the standard Euclidean scalar product; an inner product can use other positive-definite forms or act on function spaces.
• Bilinear form - Is linear in each argument but need not be symmetric or positive definite.
• Indefinite scalar product - Allows nonzero vectors with zero or negative self-product, unlike a positive-definite inner product.
• Cross product - The ordinary three-dimensional cross product returns a vector perpendicular to its inputs rather than a scalar.
• Norm - Assigns a nonnegative size to one vector; a norm arises from an inner product exactly when it satisfies the parallelogram law.

1. Which of these neighbouring kinds and how to tell them apart hold for the sense of scalar product this model covers, and on what evidence? provenance