## Introduction to the theory of bases |

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### Contents

Linear Transformations | 1 |

I1 Convergence of Series in Banach Spaces | 18 |

II1 Bases for Banach Spaces | 28 |

Copyright | |

8 other sections not shown

### Common terms and phrases

absolutely convergent Acad AF(D Amer arbitrary Banach space bases in Banach basis for H basis for X biorthogonal sequence biorthogonal system bounded boundedly complete Cauchy sequence closed linear subspace coefficient functionals compact conjugate space continuous linear converges weakly Corollary defined Definition Doklady Akad dual generalized basis element equivalent exists F-space finite dimensional follows Hence Hilbert space hypothesis implies increasing sequence inequality infinite isometric isomorphism Lemma linear topological space linear transformation locally convex Lp(S Markushevich basis metric space monotone Moreover n-ring Nauk SSSR N. S. normed linear space one-to-one orthogonal Proof proper 7t-ring reflexive retro-basis Russian Schauder basis Schauder decomposition separable Banach space shrinking sp{x Studia Math subset Theorem topological space topologically isomorphic translation-invariant metric unconditional basis unconditional convergence unconditionally convergent unique unit ball vector space weak topology weakly sequentially complete x*eX xx,fx