## Linear Operators: General theory |

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Page 67

A closed linear manifold in a

closed linear manifold in the

equation £(x*)y = x*y, y e 3). Clearly \£(x*)\ ^ so that f : X* Let rj : y** -+ x** be

defined by ...

A closed linear manifold in a

**reflexive**B-space is**reflexive**. Proof. Let 3) be aclosed linear manifold in the

**reflexive**space X. Let f : x* -h/* be defined by theequation £(x*)y = x*y, y e 3). Clearly \£(x*)\ ^ so that f : X* Let rj : y** -+ x** be

defined by ...

Page 72

B-space and 3 's a closed subspace of X, show that 3"1""1" = 3- Is the result true if

X is not

show, using Exercise 18, that both 3 a°d X/3 are

B-space and 3 's a closed subspace of X, show that 3"1""1" = 3- Is the result true if

X is not

**reflexive**? 19 If X is a**reflexive**fi-space, and 3 >s a closed subspace of X,show, using Exercise 18, that both 3 a°d X/3 are

**reflexive**. 20 If X is a B-space, ...Page 88

He used the term regular to describe what we, following Lorch [2], have called

necessary and sufficient condition for rcflcxivity is given in Theorem V.4.7.

He used the term regular to describe what we, following Lorch [2], have called

**reflexive**. Examples of**reflexive**#-spaccs are given in Chapter IV, and anecessary and sufficient condition for rcflcxivity is given in Theorem V.4.7.

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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a-finite Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space valued function vector space weak topology weakly compact weakly sequentially compact zero