Abstarct: In this paper, we investigate a series of successive and repetitive processes that lead to the best approximation, we called this process “the best approximation algorithms”, First, we describe algorithms that could find best approximation of a continuous two variable function as the sum of two continuous functions. Next we introduce some algorithms in Hilbert space and finite product of Hilbert spaces and we can assist to find an approximation for all points in Hilbert space. Finally we propose the best approximation algorithm for closed convex sets in Hilbert space using hyperplanes.