Hamelbases

A Hamel basis of a vector space V over a field F is a subset B of V that is linearly independent and spans V, meaning every element of V can be written uniquely as a finite linear combination of elements of B with coefficients in F. This definition emphasizes the algebraic notion of a basis, in contrast to topological bases that rely on infinite series.

Existence and basic properties: Every vector space has a Hamel basis provided the axiom of choice. Finite-dimensional spaces have bases consisting of a finite number of vectors, and standard linear-algebra constructions yield them. In infinite-dimensional spaces, Hamel bases can be highly nonconstructive; no explicit description of the entire basis is generally possible.

Real numbers as a vector space over the rationals: When the field F is the rationals and V is the real numbers R, a Hamel basis B exists such that every real number can be expressed uniquely as a finite Q-linear combination of elements of B. The basis is guaranteed by the axiom of choice, but no explicit recipe for its elements is known. The dimension of R over Q—that is, the cardinality of a Hamel basis for R over Q—is the cardinality of the continuum.

Cardinality considerations and contrasts: Since Q is countable, a Hamel basis for R over Q must be uncountable; in fact, its cardinality equals 2^{aleph_0}. This implies that most elements of R cannot be described constructively in terms of a simple, explicit basis. By comparison, a Schauder basis in a topological sense (used in analysis) allows infinite series, whereas a Hamel basis uses only finite sums.

Applications and context: Hamel bases are fundamental in understanding algebraic structure and dimension, and they illustrate distinctions between algebraic and topological notions of basis. They are rarely used for explicit calculations due to their nonconstructive nature.