## Summability Through Functional AnalysisSummability is an extremely fruitful area for the application of functional analysis; this volume could be used as a source for such applications. Those parts of summability which only have ``hard'' (classical) proofs are omitted; the theorems given all have ``soft'' (functional analytic) proofs. |

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### Contents

1 | |

21 | |

CHAPTER 3 TRIANGLES AND BANACH SPACE | 39 |

CHAPTER 4 FK SPACES | 51 |

CHAPTER 5 REPLACEABILITY AND CONSISTENCY | 75 |

CHAPTER 6 BIGNESS THEOREMS | 89 |

CHAPTER 7 SEQUENCE SPACES | 103 |

CHAPTER 8 INCLUSION AND MAPPING | 115 |

CHAPTER 12 DISTINGUISHED SUBSPACES OF MATRIX DOMAINS | 187 |

CHAPTER 13 DISTINGUISHED SUBSPACES OF cA | 201 |

CHAPTER 14 THE FUNCTIONAL μ | 221 |

CHAPTER 15 THE SUBSPACE P | 233 |

CHAPTER 16 SEQUENTIAL COMPLETENESS AND SEPARABILITY | 251 |

CHAPTER 17 MAPS OF BANACH SPACES | 263 |

CHAPTER 18 ALGEBRA | 281 |

CHAPTER 19 MISCELLANY | 291 |

CHAPTER 9 SEMICONSERVATIVE SPACES AND MATRICES | 141 |

CHAPTER 10 DISTINGUISHED SUBSPACES OF FK SPACES | 157 |

CHAPTER 11 EXTENSION | 179 |

303 | |

309 | |

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### Common terms and phrases

absolutely convex Banach algebra Banach space bounded sequences bounded set Cauchy Cauchy sequence CHAPTER closed graph closed subspace closure condition conservative FK space conull convergent columns coregular coregular matrix coregular mod Corollary defined DEFINITION dense dual eacts equipotent equivalent exists finite FK program FK space Fréchet space functional analysis given Hahn–Banach theorem Hahn–Banach theorem 3.0 Hausdorff hence hypothesis implies includes invariant kth column Lemma Let f lima limax locally convex locally convex space Mercerian monotone norm multiplicative Necessity notation of Remark proved regular triangle relative topology replaceable result follows row function row-finite seminorms shows subset Sufficiency summability Suppose tA)x Tauberian Theorem 12 Theorem 9 trivial u-continuous u-un u-unique vector space weak topology weakly