The standard definition of the basis for a vector space is that all the vectors can be defined as finite linear combinations of basis elements. Consider the vector space consisting of all sequences of field elements. Basis vectors could be defined as vectors which are zero except for one term in the sequence, where that term = 1. However if only finite combinations are considered then only vectors with a finite number of non-zero components would be considered as part of the space. How can a basis be defined so that all the sequences are in the space? Do we need a different definition for basis?