On the totality of sequences in topological vector spaces
Copyright policy )On the totality of sequences in topological vector spaces
AbstractPrice and Zink [Ann. of Math. 82 (1965), 139–145] gave necessary and sufficient conditions for the existence of a multiplier m so that {mφn}1∞ is total; that is, the linear span is dense in L2[0, 1], thus answering a question raised by Boas and Pollard [Bull. Amer. Math. Soc. 54 (1948), 512–522]. Using techniques similar to those of Price and Zink, it is shown that this result can be extended to more general spaces. Indeed, if X is either a separable Fréchet space or a complete separable p-normed space (0 < p ⩽ 1), then the existence of a continuous linear operator A so that {Aφn}1∞ spans a dense subspace is implied by the existence of a nested, equicontinuous family of commuting projections which in addition has the properties that the union of their ranges is dense and that, for each projection, the projection of the original sequence is total in the projected space. Conversely, in a Banach space, it is shown that if such an operator exists and is 1-1 and scalar, then such a family of projections also exists. Further, it is shown that the above considerations extend the first half of the Price-Zink result to Lp[0, 1] (0 < p < ∞) and the other half to Lp[0, 1] (1 ⩽ p < ∞).
Locally convex Fréchet spaces and (DF)-spaces, Applied Mathematics, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, Analysis, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Locally convex Fréchet spaces and (DF)-spaces, Applied Mathematics, Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces, Analysis, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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