Editing Basis (linear algebra) (section)

==== Example ====
In the study of [[Fourier series]], one learns that the functions {{math|1={1} ∪ { sin(''nx''), cos(''nx'') : ''n'' = 1, 2, 3, ... }<nowiki/>}} are an "orthogonal basis" of the (real or complex) vector space of all (real or complex valued) functions on the interval [0, 2π] that are square-integrable on this interval, i.e., functions ''f'' satisfying
<math display="block">\int_0^{2\pi} \left|f(x)\right|^2\,dx < \infty.</math>

The functions {{math|1={1} ∪ { sin(''nx''), cos(''nx'') : ''n'' = 1, 2, 3, ... }<nowiki/>}} are linearly independent, and every function ''f'' that is square-integrable on [0, 2π] is an "infinite linear combination" of them, in the sense that
<math display="block">\lim_{n\to\infty} \int_0^{2\pi} \biggl|a_0 + \sum_{k=1}^n \left(a_k\cos\left(kx\right)+b_k\sin\left(kx\right)\right)-f(x)\biggr|^2 dx = 0</math>

for suitable (real or complex) coefficients ''a''<sub>''k''</sub>, ''b''<sub>''k''</sub>. But many<ref>Note that one cannot say "most" because the cardinalities of the two sets (functions that can and cannot be represented with a finite number of basis functions) are the same.</ref> square-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 typically not useful, whereas [[orthonormal bases]] of these spaces are essential in [[Fourier analysis]].