Inner product

An inner product is a bilinear, symmetric, positive-definite function that assigns a real number to each pair of vectors. It defines notions of length and angle in a vector space.

Vector space: A set with addition and scalar multiplication satisfying the vector space axioms. Inner products are defined on vector spaces and interact with their linear structure.
Cauchy–Schwarz inequality: An inequality bounding the absolute inner product of two vectors by the product of their lengths. It makes the normalized inner product a valid cosine and underpins norm estimates.
Least squares: A method for estimating parameters by minimizing the sum of squared residuals. The normal equations express the residual's orthogonality to the model subspace.
Indefinite inner product space: A vector space equipped with a nondegenerate symmetric form that can assign negative squared lengths. It drops positivity, so self-products no longer define ordinary lengths.
Bilinear form: A function of two vector arguments that is linear in each argument separately. Bilinearity is one of the defining requirements for a real inner product.
Polarization identity: An identity recovering an inner product from the norm it induces. It shows that the induced norm retains enough information to reconstruct the real inner product.
Fourier series: A representation of a periodic function as a sum of sinusoidal modes. Orthogonality of sine and cosine modes lets inner products extract Fourier coefficients.
Semidefinite form: A symmetric bilinear form whose self-pairings are nonnegative but may vanish for nonzero vectors. It lacks the strict positivity required to make every nonzero vector have positive squared length.
Positive-definite function: A function whose associated quadratic values are nonnegative, with equality only in the specified zero case. Positive definiteness ensures a vector has zero squared length only when it is zero.
Orthogonal projection: The map sending a vector to its closest point in a subspace under an inner-product norm. Inner products characterize the residual as orthogonal to the projected subspace.

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