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\(g\)-bases in Hilbert spaces. (English) Zbl 1266.46018

Summary: The concept of \(g\)-bases in Hilbert spaces is introduced, which generalizes Schauder bases in Hilbert spaces. Some results about \(g\)-bases are proved. In particular, we characterize the \(g\)-bases and \(g\)-orthonormal bases. Dual \(g\)-bases are also discussed. We also consider the equivalent relations of \(g\)-bases and \(g\)-orthonormal bases. The property of \(g\)-minimality of \(g\)-bases is studied as well. Our results show that, in some cases, \(g\)-bases share many useful properties of Schauder bases in Hilbert spaces.

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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