When applied to

vector spaces in

mathematics,

Hamel basis means the same thing as ordinary or algebraic

basis. That is, a

Hamel basis of a

vector space V is a

linearly independent set B of

elements of V, such that every

element of V is a

finite linear combination of

elements of B.

The term Hamel basis emphasizes that, even if V is infinite-dimensional, we insist that every element of V be a *finite* linear combination of elements of B. When dealing with infinite-dimensional vector spaces, for instance in functional analysis, one frequently calls a set B a basis for V when the set of all finite linear combinations of elements of B is merely dense in V. For applications to analysis this approximation property is more natural. See orthonormal basis.