## Linear Operators: Spectral theory |

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Page 1000

If { In } were known to be uniformly convergent in a neighborhood of U , the

analyticity of its limit fy would be

sequence In is uniformly convergent on any region containing an interval of the

real axis and ...

If { In } were known to be uniformly convergent in a neighborhood of U , the

analyticity of its limit fy would be

**clear**. Unfortunately it is not**clear**that thesequence In is uniformly convergent on any region containing an interval of the

real axis and ...

Page 1064

But it is

, if f is in Co ( I ) and vanishes outside a bounded set . Next let f be in L , ( E " ) ,

and let { Im } be a sequence of functions in C° ( En ) , each vanishing outside a ...

But it is

**clear**that 8m ( x ) + g ( x ) for all x . Hence g = g , proving that go $ 1 , 111, if f is in Co ( I ) and vanishes outside a bounded set . Next let f be in L , ( E " ) ,

and let { Im } be a sequence of functions in C° ( En ) , each vanishing outside a ...

Page 1652

Then , since | F ) 2 [ F , for each F in H ( * ) ( I ) , it is

some F in LZ ( I ) . Similarly , since [ Fl « 212F12 for each F in H ( * ) ( I ) and each

index J such that \ JI Sk , it is

Then , since | F ) 2 [ F , for each F in H ( * ) ( I ) , it is

**clear**that { F } converges tosome F in LZ ( I ) . Similarly , since [ Fl « 212F12 for each F in H ( * ) ( I ) and each

index J such that \ JI Sk , it is

**clear**that if lJ Sk , the sequence ...### What people are saying - Write a review

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

46 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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 consider 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 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