Some convergence problems for weak norms

Daugavet, I.

doi:10.1090/S1061-0022-04-00823-4

Some convergence problems for weak norms
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by I. K. Daugavet
Translated by: A. Plotkin

St. Petersburg Math. J. 15 (2004), 575-585

DOI: https://doi.org/10.1090/S1061-0022-04-00823-4

Published electronically: July 6, 2004

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Abstract:

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 \[ U^{(n)}=\{ u\in U \mid \chi (u)=0, \chi \in U_n^*\}. \] If the sequence $\{U_n^*\}$ is asymptotically dense in $U^*$, then $\|I_n\|\to 0$, 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 $\|u-P_nu\|_V/\|u-P_nu\|_U\to 0$. The norm $\|I_n\|$ 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.

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