Hamelbasis

A Hamel basis, named after David Hilbert's work in functional analysis, is a basis of a vector space that uses finite linear combinations. Formally, a subset B of a vector space V over a field F is a Hamel basis if every element v in V can be written uniquely as a finite sum v = a1 b1 + … + an bn with coefficients ai in F and basis elements bi in B. In this sense, B is linearly independent and spans V, and the representation of each vector uses only finitely many basis vectors.

Existence and size: Every vector space has a Hamel basis, but the proof relies on the axiom of choice (via Zorn’s lemma) and provides a nonconstructive existence result. Finite-dimensional spaces over F have finite Hamel bases, with dimension equal to the usual dimension. Infinite-dimensional spaces have infinite Hamel bases; any two Hamel bases of a given space have the same cardinality, defining the Hamel dimension of the space.

Examples: In R^n, the standard basis e1, …, en is a Hamel basis. For the space of polynomials with real coefficients, {1, x, x^2, …} is a Hamel basis. For the real numbers R viewed as a vector space over the rational numbers Q, there exists a Hamel basis B such that every real number is a finite Q-linear combination of elements of B; such a basis has cardinality equal to the continuum, though no explicit description is known.

Relation to other bases: A Hamel basis uses finite linear combinations and is purely algebraic. It is distinct from a Schauder basis, which pertains to topological vector spaces and allows infinite (convergent) series representations. The concept is essential for defining the dimension of a vector space in the absence of topology.