Abstract
Throughout this chapter (and the next two) the basic space X will be a Banach space. According to the three most common used topologies, there are bases for the strong, the weak and the weak* topologies for X, whose definitions are given in the first paragraph. It is shown that every basis for X is a Schauder basis, a basis with continuous linear coefficient functionals. The next paragraph shows under which conditions a biorthogonal system is a basis for X, the equivalence of strong and weak Schauder bases for X and relations between bases for X and bases for the adjoint space X*. Three paragraphs are devoted to retro-, shrinking, boundedly complete, unconditional, absolutely convergent and uniform bases. Some applications of summability methods on the theory of bases are given in the sixth section and in the last paragraph bases for the special spaces c0, lp(l ≤ p < ∞), C[0,1], A L[0,l] (l ≤ p < ∞), L2[0,2π] and A 2 are considered.
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© 1969 Springer-Verlag Berlin · Heidelberg
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Marti, J.T. (1969). Bases for Banach Spaces. 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_3
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DOI: https://doi.org/10.1007/978-3-642-87140-5_3
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