## Linear Operators: General theory |

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Page 311

Theorem 5.1 shows that there is an

S, Z) and ba(S, Z), which is determined by the equation x*%E -— /u.(E), E e Z.

Thus, since B(S,Z) is equivalent to C^), ba(S,Z) is equivalent to ra^iSj) (Theorem ...

Theorem 5.1 shows that there is an

**isometric isomorphism**x* « — - /t between B*(S, Z) and ba(S, Z), which is determined by the equation x*%E -— /u.(E), E e Z.

Thus, since B(S,Z) is equivalent to C^), ba(S,Z) is equivalent to ra^iSj) (Theorem ...

Page 313

The correspondence U -.fa -> faisan

Sv E2). (c) // Ex is in Ex then v(/iv Ex) = v(U(fa), £j) for all fa in ba(Sv Proof.

Recalling that x is an isomorphism of E onto Ev it is clear that the mapping T is an

...

The correspondence U -.fa -> faisan

**isometric isomorphism**of ba(Sv Ey) onto ca(Sv E2). (c) // Ex is in Ex then v(/iv Ex) = v(U(fa), £j) for all fa in ba(Sv Proof.

Recalling that x is an isomorphism of E onto Ev it is clear that the mapping T is an

...

Page 337

Thus, ba(S, E) is isometrically isomorphic with the closed subspace BV0(I) of all /

e BV(I) such that /(a + ) = 0. If N is the ... Thus / □* — »• /xt determines an

obtain the ...

Thus, ba(S, E) is isometrically isomorphic with the closed subspace BV0(I) of all /

e BV(I) such that /(a + ) = 0. If N is the ... Thus / □* — »• /xt determines an

**isometric isomorphism**between NBV(I) and rba(I, E). Using Theorem 9.9, weobtain the ...

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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