Davit Karagulyan: Faber–Schauder and Franklin systems

We will start by giving a criterion for a sequence of functions to be a basis (an unconditional basis) in a Banach space. Next we will construct an example of a basis in C[0,1] called Faber-Schauder system. Applying the Gram-Schmidt orthogonalization process on that system we will obtain an orthonormal system in C[0,1], which is called Franklin system. Then we will discuss its basisness properties in spaces C[0,1] and L^p.
At the end I will say few words about Kolmogorov's rearrangement conjecture.