Let \(Q_ i\), \(i= 1,\dots, m\), be a finite family of closed convex sets in a Hilbert space \(H\) with a nonempty intersection. The convex feasibility problem (CFP) is to find an element \(x^*\in Q= \bigcap^ m_{i= 1} Q_ i\). The block-iterative projection algorithmic scheme for solving the CFP problem in \(n\)-dimensional Euclidean space iteratively generates the following sequence. Choose an initial point \(x^ 0\in H\) and, for each \(k\) set \(x^{k+ 1}= x^ k+ \lambda_ k \sum^ m_{i= 1} w_ k(i) (P_ i(x^ k)- x^ k)\), where \(P_ i(x^ k)\) is the orthogonal projection of \(x^ k\) onto the set \(Q_ i\), \(w_ k: \{1,\dots, m\}\to \mathbb{R}_ +\) is a weight function \((\sum^ m_{i= 1} w_ k(i)= 1)\) and \(\lambda_ k \in\mathbb{R}_ +\) are relaxation parameters, \(0< \varepsilon_ 1\leq \lambda_ k\leq \varepsilon_ 2< 2\). The objective of the paper is to study the generalization of this method to the problem in a Hilbert space \(H\) and the convergence of this generalization. It is known that under quite mild conditions on the weight functions and regardless of the choice of the initial parameters this method generates a weakly convergent sequence regardless of the choice of the initial point. It is shown that ensuring the strong convergence of such sequence is more difficult and usually demands some additional conditions on the set of the \(Q_ i\) themselves.