Hamel basis

Analysis

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 (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 ${\displaystyle c_{00}}$, the space of the sequences${\displaystyle x=(x_{n})}$ of real numbers that have only finitely many non-zero elements, with the norm ${\textstyle \|x\|=\sup _{n}|x_{n}|}$. Its standard basis, consisting of the sequences having only one non-zero element, which is equal to 1, is a countable Hamel basis.