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The concept of g-basis in the Hilbert spaces is introduced by Guo (2012) who generalizes the Schauder basis in the Hilbert spaces. g-basis plays the similar role in g-frame theory to that the Schauder basis plays in frame theory. In this paper, we establish some important properties of g-bases in the Hilbert spaces. In particular, we obtain a simple condition under which some important properties established in Guo (2012) are still true. With these conditions, we also establish some new interesting properties of g-bases which are related to g-minimality. Finally, we obtain a perturbation result about g-bases.

Some New Characterizations and g-Minimality and Stability of g-Bases in the Hilbert Spaces