On the theory of fundamental norming bounded biorthogonal systems in Banach spaces

Abstract:

Let $X$ and $Y$ be quasi complementary subspaces of a separable Banach space $B$ and let $({z_n})$ be a sequence complete in $X$. Then (a) there exists a uniformly minimal norming $M$-basis $({x_n})$ of $X$ with ${x_m} \in \operatorname {span} {({z_n})_{n \geqslant {q_m}}}$ for every $m$, ${q_m} \to \infty$; (b) if $({x_n})$ is a uniformly minimal norming $M$-basis of $X$, there exists a uniformly minimal norming $M$-basis of $B$ which is an extension of $({x_n})$; (c) there exists a uniformly minimal norming $M$-basis $({x_n}) \cup ({y_n})$ of $B$ with $({x_n}) \subset X$ and $({y_n}) \subset Y$.