## Linear Operators: General theory |

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Page 72

16 Let X be a normed linear space which is not assumed to be complete, and let

3 be a closed

17 Definition'. If X is a normed linear space, and ZQ2L, the set Z1- = {x*\x* e X*, ...

16 Let X be a normed linear space which is not assumed to be complete, and let

3 be a closed

**subspace**of X. Then, if 3 and X/8 arc complete, show that X is also.17 Definition'. If X is a normed linear space, and ZQ2L, the set Z1- = {x*\x* e X*, ...

Page 436

1.7) set, then X must be finite dimensional. 6 Let X be a fi-space, and Xx a

topology of Xp 7 Let X be a linear space, and r a total

a set A Q X ...

1.7) set, then X must be finite dimensional. 6 Let X be a fi-space, and Xx a

**subspace**of X. Show that the Xf topology of Xj is the same as the relative X*topology of Xp 7 Let X be a linear space, and r a total

**subspace**of X+. Show thata set A Q X ...

Page 513

Let T : X -*□ Q be a bounded linear operator mapping into a second B-space.

Then T(X) is closed if and only if T(^j) is closed. 18 Every non-null, finite

dimensional, proper

mapping onto it.

Let T : X -*□ Q be a bounded linear operator mapping into a second B-space.

Then T(X) is closed if and only if T(^j) is closed. 18 Every non-null, finite

dimensional, proper

**subspace**of a B-space has infinitely many projectionsmapping onto it.

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero