Abstract
It is quite natural to generalize the concept of a basis for a space X by taking a sequence of linear (not necessarily closed) subspaces of X instead of a sequence of elements of X and, in the same time, to define X to be an F-space. Such a basis of subspaces is called a decomposition and, if the subspaces are closed, a Schauder decomposition. The closure of the subspaces is intimately connected with the continuity of projections onto these subspaces. If X is a Banach space, it turns out that X always has a decomposition, but there are (non-separable) Banach spaces which do not have a Schauder decomposition. Concerning the existence of Schauder decompositions of Banach spaces, a theorem is verified which corresponds to the theorem of Nikol’skiĭ for the existence of a basis. Finally, it is shown that the notions of a weak Schauder decomposition and a Schauder decomposition in a Banach space are equivalent.
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Marti, J.T. (1969). Decompositions. In: Introduction to the Theory of Bases. Springer Tracts in Natural Philosophy, vol 18. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-87140-5_7
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DOI: https://doi.org/10.1007/978-3-642-87140-5_7
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