Successive projections on hyperplanes (Q1058267)

Let H be a family of linear manifolds in n-space. Assume that there is a finite number of linear spaces \(\{V_ i\}\) such that each element of H is a translate of some \(V_ i\) and that the distance from the origin is bounded. The main result of the paper states that any sequence of projections onto manifolds from the family H is bounded. Specifically, let \(x_{k+1}=p_ k(x_ k)\), \(k=0,1,..\). where \(p_ k\) is the projection on some manifold of the family H, then \(\sup_{k}| x_ k| \leq | x_ 0| +r\), where r depends only on H but not on the choice of the sequence \((p_ k)\).