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Hamel basis

Basic definition

In mathematics, a Hamel basis of a vector space is a set B of vectors in the space such that

See also basis (linear algebra).

An "orthonormal basis" need not be a Hamel basis

In the study of Fourier series, one learns that the functions { 1} ∪ { sin(nx), cos(nx) : n = 1, 2, 3, ... } are an "orthonormal basis" of the set of all complex-valued functions that are quadratically integrable on the interval [0, 2π], i.e., functions f satisfying

These functions are linearly independent, and every function that is quadratically integrable on that interval is an "infinite linear combination" of them. That means that

for suitable coefficients ak, bk. But most quadratically integrable functions cannot be represented as finite linear combinations of these basis functions, which therefore do not comprise a Hamel basis. Every Hamel basis of this space is much bigger than this merely countably infinite set of functions. Hamel bases of spaces of this kind are of little if any interest; orthonormal bases of these spaces are important to Fourier analysis.