## 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 ZCX, the set Z1- = {x*\x* € 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/3 are complete, show that X is also.17 Definition. If X is a normed linear space, and ZCX, the set Z1- = {x*\x* € X*, ...

Page 436

6 Let X be a B-space, and Xj a

the same as the relative X* topology of Xv 7 Let X be a linear space, and r a total

6 Let X be a B-space, and Xj a

**subspace**of X. Show that the X* topology of Xx isthe same as the relative X* topology of Xv 7 Let X be a linear space, and r a total

**subspace**of X+. Show that a set A Q X is /'-bounded if and only if f(A) is a ...Page 513

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

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

dimensional, proper

**subspace**of a Zf-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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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional linear map linear operator linear topological space LP(S measurable functions 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 Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact