Bessel's inequality

Bessel's inequality states that, for an orthonormal sequence in an inner product space, the sum of squared inner products with any vector does not exceed the vector’s squared norm.

Orthogonal projection: The component of a vector lying in a chosen subspace, with the residual perpendicular to that subspace. Projecting onto finitely many orthonormal vectors yields the norm bound directly.
Inner product space: A vector space equipped with an inner product that defines lengths and angles. Bessel’s inequality is formulated through the inner products and norms of this structure.
Fourier series: A representation of a periodic function as a sum of sinusoidal modes. Bessel’s inequality bounds the energy in any selected collection of Fourier coefficients.
Fourier analysis: The study of representing functions and signals through their frequency components. Bessel’s inequality is a basic energy estimate across Fourier coefficients.
Pythagorean theorem: A geometric relation stating that squared lengths of perpendicular sides sum to the squared length of their hypotenuse. The inequality follows by applying this relation to a vector and its orthogonal projection.
Orthonormal set: A set of unit vectors that are pairwise orthogonal under an inner product. The inequality’s coefficient sum is formed from inner products with these vectors.
Fourier transform: An operation expressing a function in terms of its frequency components. Related energy bounds constrain frequency-domain coefficients and transforms.
Riesz–Fischer theorem: A theorem connecting square-summable coefficient sequences with functions in the relevant complete function space. It complements coefficient bounds by establishing when coefficient data can define a function.
Gram matrix: A matrix of pairwise inner products among a finite list of vectors. Its positive semidefiniteness gives a finite-dimensional algebraic route to the inequality.
Norm (mathematics): A function assigning lengths to vectors while satisfying positivity, homogeneity, and the triangle inequality. The right-hand side is the square of the vector’s norm.

Linked from 6 pages