GRAM-SCHMIDT PROCESS OF ORTHONORMALIZATION IN BANACH SPACES

Abstract

Gram-Schmidt orthonormalization in Banach spaces is considered. Using this orthonormalization process we can prove that if $P$ is a projection on a reflexive Banach space $X$ with a basis $\{e_n;f_n\}$, then there exists a basis $\{u_n;g_n\}$ of $X$ such that $\{g_n\}\approx\{f_n\}$ and the matrix of $P$ with respect to $\{u_n;g_n\}$ has the property that all but a finite number of entries of each column and each row are zero.