An infinite sequence of vectors {v_n} in a Banach space such that every v ε‎V can be written uniquely as $v={\sum }_1^{\infty }{a}_n{v}_n$ for scalars a_n. Every Banach space with a Schauder basis is necessarily separable, but the converse does not hold. Compare baire Category Theorem, complete orthonormal basis, Hamel basis.