Abstract
A subset M of a separable Hilbert space H is ℓ1-bounded if there exists a Riesz basis F={en}n∈N for H such that supx∈M∑n∈N|〈x,en〉|<∞. A similar definition for ℓ1-frame-bounded sets is made by replacing Riesz bases with frames. This paper derives properties of ℓ1-bounded sets, operations on the collection of ℓ1-bounded sets, and the relation between ℓ1-boundedness and ℓ1-frame-boundedness. Some open problems are stated, several of which have intriguing implications.
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