  Let $L_1,L_2,\dots,L_K$ be a family of closed subspaces of a Hilbert space
$H$, $L_1\cap \dots \cap L_K =\{0\}$; let $P_k$ be the orthogonal projection
onto $L_k$. We consider two types of consecutive projections of an element
$x_0\in H$: alternating projections $T^nx_0$, where $T=P_K\circ\dots\circ P_1$,
and remotest projections $x_n$ defined recursively, $x_{n+1}$ being the
remotest point for $x_n$ among $P_1x_n,\dots,P_Kx_n$. These $x_n$ can be
interpreted as residuals in greedy approximation with respect to a special
dictionary associated with $L_1,L_2,\dots,L_K$. We establish parallels between
convergence properties separately known for alternating projections, remotest
projections, and greedy approximation in $H$.
  Here are some results. If $L_1^\perp+\dots+L_K^\perp=H$, then $x_n\to 0$
exponentially fast. In case $L_1^\perp+\dots+L_K^\perp\not=H$, the convergence
$x_n\to 0$ can be arbitrarily slow for certain $x_0$. Such a dichotomy,
exponential rate of convergence everywhere on $H$, or arbitrarily slow
convergence for certain starting elements, is valid for greedy approximation
with respect to general dictionaries. The dichotomy was known for alternating
projections. Using the methods developed for greedy approximation we prove that
$|T^nx_0|\le C(x_0,K)n^{-\alpha(K)}$ for certain positive $\alpha(K)$ and all
starting points $x_0\in L_1^\perp+\dots+L_K^\perp$.
