Hamel dimension

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a Hamel basis of V. It is sometimes called Hamel dimension when it is necessary to distinguish it from other types of dimension. Every basis of a vector space has equal cardinality and so the Hamel dimension of a vector space is uniquely defined.

The Hamel dimension is a natural generalization of the dimension of Euclidean space, since Eⁿ is a vector space of dimension n over R (the reals). However, the Hamel dimension depends on the base field, so while R has dimension 1 when considered as a vector space over itself, it has dimension c (the cardinality of the continuum) when considered as a vector space over Q (the rationals).

Some simple formulae relate the Hamel dimension of a vector space with the cardinality of the base field and the cardinality of the space itself. If V is a vector space over a field K then, denoting the Hamel dimension of V by dimV, we have:

If dimV is finite, then |V| = |K|^dimV.

If dimV is infinite, then |V| = max(|K|, dimV).