of a basis for Lp([0, 1]), when 1 ≤ p < ∞. When , another example is the trigonometric system defined below. The Banach space C([0, 1]) of continuous functions on the interval [0, 1], with the supremum norm, admits a Schauder basis. The Faber–Schauder system is the most commonly used Schauder basis for C([0, 1]). Several bases for classical spaces were discovered before Banach's book appeared (), but some other cases remained open for a long time. For example, the question of whether the disk algebra A(D) has a Schauder basis remained open for more than forty years, until Bočkarev showed in 1974 that a basis constructed from the Franklin system exists in A(D). One can also prove that the periodic Franklin system is a basis for a Banach space Ar isomorphic to A(D). This space Ar consists of all complex continuous functions on the unit circle T whose conjugate function is also continuous. The Franklin system is another Schauder basis for C([0, 1]), and it is a Schauder basis in Lp([0, 1]) when . Systems derived from the Franklin system give bases in the space C1([0, 1]2) of differentiable functions on the unit square. The existence of a Schauder basis in C1([0, 1]2) was a question from Banach's book. Relation to Fourier series Let {xn} be, in the real case, the sequence of functions or, in the complex case, The sequence {xn} is called the trigonometric system. It is a Schauder basis for the space Lp([0, 2π]) for any p such that . For p = 2, this is the content of the Riesz–Fischer theorem, and for p ≠ 2, it is a consequence of the boundedness on the space Lp([0, 2π]) of the Hilbert transform on the circle. It follows from this boundedness that the projections PN defined by are uniformly bounded on Lp([0, 2π]) when . This family of maps {PN} is equicontinuous and tends to the identity on the dense subset consisting of trigonometric polynomials. It follows that PNf tends to f in Lp-norm for every . In other words, {xn} is a Schauder basis of Lp([0, 2π]). However, the set {xn} is not a Schauder basis for L1([0, 2π]). This means that there are functions in L1 whose Fourier series does not converge in the L1 norm, or equivalently, that the projections PN are not uniformly bounded in L1-norm. Also, the set {xn} is