Orthonormal basis

An orthonormal basis is an orthonormal family in a Hilbert space whose finite linear combinations are dense in that space. Every vector is determined by its coefficients along the family.

Hilbert space: A complete inner-product space in which limits of Cauchy sequences remain in the space. Orthonormal bases are defined for Hilbert spaces, where completeness supports convergent expansions.
Fourier series: A representation of a periodic function as a sum of sine and cosine waves, or complex exponentials. Trigonometric orthonormal bases turn functions into convergent sums of frequency components.
L2 space: The Hilbert space of square-integrable functions, with functions identified when equal almost everywhere. Its trigonometric and other orthonormal bases support function approximation and signal analysis.
Orthonormal basis (linear algebra): A finite orthonormal vector family that spans a finite-dimensional inner-product space. The finite-dimensional meaning requires exact spanning, unlike density of finite spans in infinite dimensions.
Inner product: A function that assigns scalar products to pairs of vectors and induces length and angle. It defines both vector norms and the orthogonality required of the family.
Fourier coefficient: A scalar measuring a function's component along a specified Fourier basis function. Inner products with basis vectors provide the coefficients of an expansion.
Discrete Fourier transform: A finite transform that expresses a sequence as a sum of discrete complex-frequency components. It computes coordinates in a finite Fourier basis for sampled data.
Schauder basis: A sequence in a Banach space giving every vector a unique convergent ordered series expansion. A Schauder basis need not be orthogonal, while an orthonormal basis has inner-product structure.
Orthogonality: A relation in an inner-product space in which two vectors have inner product zero. Distinct basis vectors must be pairwise orthogonal.
Orthogonal projection: The nearest-point projection of a vector onto a closed subspace of an inner-product space. Finite-dimensional projections onto basis spans converge to the represented vector.

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