When applied to vector space
s 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 element
s of V, such that every element
of V is a finite linear combination
s 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.