Hamel basis

Basic definition

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

• The set B is linearly independent. That means that no linear combination of finitely many members of B is 0, except the trivial linear combination in which all coefficients are 0. Some mathematicians may be quick to say that the word finitely is a redundancy; that finiteness is part of the very definition of the concept of linear combination. If so, that is why redundancy is sometimes urgently needed, since the point about finiteness is easily forgotten when these concepts are applied to infinite-dimensional inner product spaces (concerning which more appears below).

• Every vector in the space can be represented as a linear combination of (just finitely many) members of B.

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 a_k, b_k. 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.