Schauder Basis

High Quality Content by WIKIPEDIA articles! The standard bases of c0 and lp for 1 ¿ p < ¿ are Schauder bases. Every orthonormal basis in a separable Hilbert space is a Schauder basis. The Haar system is an example of a basis for Lp(0, 1) with 1 ¿ p < ¿. Another example is the trigonometric system defined below. The Banach space C of continuous functions on the interval, with the supremum norm, admits a Schauder basis. A Banach space with a Schauder basis is necessarily separable, but the converse is false; that is, there exists a separable Banach space without a Schauder basis.[3] A Banach space with a Schauder basis has the approximation property. A theorem of Mazur asserts that every Banach space has an (infinite-dimensional) subspace with a basis. A question of Banach asked whether every separable Banach space has a basis; this was negatively answered by Per Enflo who constructed a Banach space without a basis.