## Linear Operators, Part 2 |

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Page 932

Let S be an abstract set and E a field ( resp . o - field ) of subsets of S . Let F be an

additive ( resp . weakly

operators on a Hilbert space H satisfying F ( 0 ) = 0 and F ( S ) = 1 . Then there

exists ...

Let S be an abstract set and E a field ( resp . o - field ) of subsets of S . Let F be an

additive ( resp . weakly

**countably additive**) function on to the set of positiveoperators on a Hilbert space H satisfying F ( 0 ) = 0 and F ( S ) = 1 . Then there

exists ...

Page 958

This argument shows that the vector valued additive set function y is weakly

theorem of Pettis ( IV . 10 . 1 ) it is

, ly ( rn ) ...

This argument shows that the vector valued additive set function y is weakly

**countably additive**on the o - field consisting of all Borel subsets of e . By atheorem of Pettis ( IV . 10 . 1 ) it is

**countably additive**in the strong topology , i . e ., ly ( rn ) ...

Page 959

Since Mo is

for some m , Moleem ) 2 Ho ( eemba ) > k - € . This shows that the set function u is

uniquely defined on B . Next we show that u is

Since Mo is

**countably additive**on Boy Molebn ) = limm Moleembn ) 2k , and sofor some m , Moleem ) 2 Ho ( eemba ) > k - € . This shows that the set function u is

uniquely defined on B . Next we show that u is

**countably additive**. For this ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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additive adjoint operator algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function give given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero