Editing Basis (linear algebra) (section)

=== Analysis ===
In the context of infinite-dimensional vector spaces over the real or complex numbers, the term '''{{visible anchor|Hamel basis}}''' (named after [[Georg Hamel]]<ref>{{Harvnb|Hamel|1905}}</ref>) or '''algebraic basis''' can be used to refer to a basis as defined in this article. This is to make a distinction with other notions of "basis" that exist when infinite-dimensional vector spaces are endowed with extra structure. The most important alternatives are [[orthogonal basis|orthogonal bases]] on [[Hilbert space]]s, [[Schauder basis|Schauder bases]], and [[Markushevich basis|Markushevich bases]] on [[normed linear space]]s. In the case of the real numbers '''R''' viewed as a vector space over the field '''Q''' of rational numbers, Hamel bases are uncountable, and have specifically the [[cardinality]] of the continuum, which is the [[cardinal number]] {{nowrap|<math>2^{\aleph_0}</math>,}} where <math>\aleph_0</math> ([[aleph-nought]]) is the smallest infinite cardinal, the cardinal of the integers.

The common feature of the other notions is that they permit the taking of infinite linear combinations of the basis vectors in order to generate the space. This, of course, requires that infinite sums are meaningfully defined on these spaces, as is the case for [[topological vector space]]s – a large class of vector spaces including e.g. [[Hilbert space]]s, [[Banach space]]s, or [[Fréchet space]]s.

The preference of other types of bases for infinite-dimensional spaces is justified by the fact that the Hamel basis becomes "too big" in Banach spaces: If ''X'' is an infinite-dimensional normed vector space that is [[complete space|complete]] (i.e. ''X'' is a [[Banach space]]), then any Hamel basis of ''X'' is necessarily [[uncountable]]. This is a consequence of the [[Baire category theorem]]. The completeness as well as infinite dimension are crucial assumptions in the previous claim. Indeed, finite-dimensional spaces have by definition finite bases and there are infinite-dimensional (''non-complete'') normed spaces that have countable Hamel bases. Consider {{nowrap|<math>c_{00}</math>,}} the space of the [[sequence]]s <math>x=(x_n)</math> of real numbers that have only finitely many non-zero elements, with the norm {{nowrap|<math display="inline">\|x\|=\sup_n |x_n|</math>.}} Its [[standard basis]], consisting of the sequences having only one non-zero element, which is equal to 1, is a countable Hamel basis.

==== 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]].