## Linear Operators, Part 2 |

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Page 1180

Therefore , Corollary 23 generalizes , with hardly any change in its proof , to the

space of functions | with values in any space L , ( Ý ) , y denoting an arbitrary

with ...

Therefore , Corollary 23 generalizes , with hardly any change in its proof , to the

space of functions | with values in any space L , ( Ý ) , y denoting an arbitrary

**Hilbert space**. Next , it may be noted that Lemma 24 generalizes at once , andwith ...

Page 1262

28 Let a self adjoint operator A in a

there exists a

that Ax = PQx , XEH , P denoting the orthogonal projection of Hi on H . 29 Let { Tn

} ...

28 Let a self adjoint operator A in a

**Hilbert space**H with O SA SI be given . Thenthere exists a

**Hilbert space**H , 2H , and an orthogonal projection Q in Hi suchthat Ax = PQx , XEH , P denoting the orthogonal projection of Hi on H . 29 Let { Tn

} ...

Page 1773

APPENDIX

numbers , together with a complex function ( • , • ) defined on HXH with the

following properties : ( i ) ( x , x ) = 0 if and only if x = 0 ; ( ii ) ( x , x ) 20 , XEH ; ( iii )

( x + y ...

APPENDIX

**Hilbert space**is a linear vector space H over the field 0 of complexnumbers , together with a complex function ( • , • ) defined on HXH with the

following properties : ( i ) ( x , x ) = 0 if and only if x = 0 ; ( ii ) ( x , x ) 20 , XEH ; ( iii )

( x + y ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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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