Representation of the space spanned by a sequence in a Banach space
A sequence \((x_ n)\) of a Banach space B is said to be pseudo basic with brackets if there exists an increasing sequence \((q_ n)\) of positive integers so that, setting \(q_ 0=0\), \(x=\sum^{\infty}_{m=0}(\sum^{q_{m+1}}_{n=q_ m+1}a_ nx_ n)\) for every x of \(\overline{span}(x_ n)\), where \((a_ n)\) depends on x while \((q_ n)\) does not depend on x; this \((x_ n)\) becomes basic with brackets if it is also minimal. It is proved that, if \(\overline{span}(x_ n)\) has infinite dimension, there exist \((f_ n)\) of \(B^*\) and \((q_ n)\) of positive increasing integers such that \[ x=\lim_{m\to \infty}(\sum^{q_ m}_{n=1}f_ n(x)x_ n)+\sum^{q_{m+1}}_{n=q_ m+1}a_ nx_ n) \] for every x of \(\overline{span}(x_ n)\), where \((a_ n)\) depends on x but \((q_ n)\) does not depend on x. In general this \((f_ n)\) is not unique and \((x_ n,f_ n)\) is biorthogonal only if \((x_ n)\) is minimal. It follows that there exist two complementary subsequences \((y_ n)\) and \((z_ n)\) of \((x_ n)\) such that, setting \(X=\overline{span}(x_ n)\), \(Y=\overline{span}(y_ n)\) and \(Z=\overline{span}(z_ n)\), \((y_ n+Z)\) and \((z_ n+Y)\) are pseudo bases with brackets of X/Z and X/Y respectively. In particular \((y_ n+Z)\) and \((z_ n+Y)\) become basic with brackets if \((x_ n)\) is minimal; moreover also \((y_ n)\) and \((z_ n)\) become basic with brackets if \((x_ n)\) is minimal and norming. These properties solve some open questions.
scientific article; zbMATH DE number 3929686