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.

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