Distance formulas on weighted Banach spaces of analytic functions

[EN] Let v be a radial weight function on the unit disc or on the complex plane. It is shown that for each analytic function f0 in the Banach space Hv all analytic functions f such that v|f| is bounded, the distance of f0 to the subspace Hv0 of Hv of all the functions g such that v|g| vanishes at infinity is attained at a function g0Hv0. Moreover a simple, direct proof of the formula of the distance of f to Hv0 due to Perfekt is presented. As a consequence the corresponding results for weighted Bloch spaces are obtained.