On the Kaczmarz algorithm of approximation in infinite-dimensional spaces

The Kaczmarz algorithm of successive projections suggests the following concept. A sequence (e(k)) of unit vectors in a Hilbert space is said to be effective if for each vector x in the space the sequence (x(n)) converges to x where (x(n)) is defined inductively: x(0) = 0 and x(n) = x(n-1) + alpha(n)e(n), where alpha(n) = [x - x(n-1), e(n)]. We prove the effectivity of some sequences in Hilbert spaces. We generalize the concept of effectivity to sequences of vectors in Banach spaces and we prove some results for this more general concept.