Hamel bases

Analysis

In the context of infinite-dimensional vector spaces over the real or complex numbers, the term Hamel basis (named after Georg Hamel) 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 bases on Hilbert spaces, Schauder bases, and Markushevich bases on normed linear spaces. In the case of the real numbers ${\displaystyle \mathbb {R} }$ viewed as a vector space over the field ${\displaystyle \mathbb {Q} }$ of rational numbers, Hamel bases are uncountable, and have specifically the cardinality of the continuum, which is the cardinal number${\displaystyle 2^{\aleph _{0}}}$, where ${\displaystyle \aleph _{0}}$ (aleph-nought) is the smallest infinite cardinal, the cardinal of the integers.