Uniformly bounded orthonormal polynomials on the sphere

Given any ε > 0, we construct an orthonormal system of nk uniformly bounded polynomials of degree at most k on the unit sphere in R where nk is bigger than 1− ε times the dimension of the space of polynomials of degree at most k. Similarly we construct an orthonormal system of sections of powers A of a positive holomorphic line bundle on a compact Kähler manifold with cardinality bigger than 1− ε times the dimension of the space of global holomorphic sections to A.