IJPAM: Volume 52, No. 1 (2009)

$X_d$-FRAMES IN BANACH SPACES AND THEIR DUALS

Diana T. Stoeva
Department of Mathematics
University of Architecture Civil Engineering and Geodesy
Blvd. Hristo Smirnenski 1, Sofia, 1046, BULGARIA
e-mail: [email protected]


Abstract.We consider consequences of the lower and the upper $X_d$-frame conditions. The lower $X_d$-frame condition is proved to be necessary for existence of some series expansions. Our main interest is on duals and dual$^*$s. We consider connection between dual and dual$^*$ of an $X_d$-Bessel sequence, and necessary and sufficient conditions for their existence. If $X_d$ has the canonical vectors as a Schauder basis, then an $X_d$-Bessel sequence, having a dual or dual$^*$, is moreover a Banach frame.

Received: July 10, 2008

AMS Subject Classification: 42C15, 46B15

Key Words and Phrases: dual, dual$^*$, $X_d$-Bessel sequence, $X_d$-frame, Banach frame, series expansions

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 52
Issue: 1