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The Basis Problem Some Properties of Bases in Banach Spaces 1 Definition of a basis in a Banach space The basis problem Relations between bases i...
2 Some examples of bases in concrete Banach spaces Some separable Banach spaces in which no basis is known
The coefficient functional associated to a basis Bounded bases Nor malized bases
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arbitrary Assume Bases in Banach basis of c0 basis of type basis xn Besselian biorthogonal system BK-space block basic sequence bounded basis Cauchy sequence completes the proof condition Consequently continuous linear mapping corollary defined definition denote dimensional Banach space element exists a constant exists a sequence finite sequences Furthermore Haar functions Hence Hilbert space Hilbertian implication increasing sequence infinite dimensional Banach initial norm isometry isomorphic lemma linear subspace linearly independent Math matrix norm monotone basis natural basis normalized basis obtain obviously orthogonal Pelczynski permutation positive integer proof of lemma proof of proposition proof of theorem properties prove Rademacher functions real Banach space remark respectively satisfying Schauder basis separable Banach space sequence of scalars sequence space statements are equivalent subsymmetric sup sup T-norm taking into account theorem 4.1 topology type wc0 unconditional basis unit vector basis weakly closed whence