## Linear Operators, Part 2 |

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Page 1000

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

analyticity of its limit fu 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 fu 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

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

let { m } be a sequence of functions in C° ( E " ) , each vanishing outside a ...

But it is

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

let { m } be a sequence of functions in C° ( E " ) , each vanishing outside a ...

Page 1652

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

some F in Lg ( 1 ) . Similarly , since ( F16 ) 210F , for each F in H ( * ) ( I ) and

each index J such that \ JI S k , it is

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

**clear**that { Fn } converges tosome F in Lg ( 1 ) . Similarly , since ( F16 ) 210F , for each F in H ( * ) ( I ) and

each index J such that \ JI S k , it is

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

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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