Some convergence problems for weak norms

Let $U$ be a normed space compactly embedded in a space $V$, let $\{U_n^*\}$ be a sequence of finite-dimensional subspaces of the dual space $U^*$, and let

If the sequence $\{U_n^*\}$ is asymptotically dense in $U^*$, then $\Vert I_n\Vert\to 0$, where $I_n$ denotes the operator that embeds $U^{(n)}$ in $V$. In particular, if $\{P_n\}$ is a sequence of finite-dimensional projections in $U$, and the sequence $\{{\mathcal R}(P_n^*)\}$ is asymptotically dense in $U^*$, then $\Vert u-P_nu\Vert _V/\Vert u-P_nu\Vert _U\to0$. The norm $\Vert I_n\Vert$ is estimated in terms of the best approximation of the elements of the unit ball in $V^*$ (this ball is compact in $U^*$) by elements of $U_n^*$. Usually, for projection methods of solving functional equations, the metric in which the convergence should be studied is dictated by the general convergence theorems (we mean, e.g., the energy metric for the Ritz method). The above arguments make it possible to establish a faster convergence of projection methods in weaker metrics. Some results of this type are obtained in the paper for the Ritz and Galerkin methods and for the method of moments.