## Linear Operators: Spectral theory |

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Page 1179

which sends a scalar - valued function with the Fourier transform | ( 5 ) into the

vector - valued function whose nth component has the Fourier transform in ( 5 )

defined ...

**PROOF**. We saw in the course of proving Theorem 25 that the mapping MXwhich sends a scalar - valued function with the Fourier transform | ( 5 ) into the

vector - valued function whose nth component has the Fourier transform in ( 5 )

defined ...

Page 1724

g ) = ( f , Sg ) for f in D ( T ) and g in D ( S ) . By Green ' s formula , proved in the

last paragraph of Section 2 , this equation is valid if | and g are in CO ( I ) . It

follows ...

**Proof**. By the preceding lemma and by Corollary 11 it suffices to show that ( T ) ,g ) = ( f , Sg ) for f in D ( T ) and g in D ( S ) . By Green ' s formula , proved in the

last paragraph of Section 2 , this equation is valid if | and g are in CO ( I ) . It

follows ...

Page 1750

We shall see , however , that this fact is needed in the course of the

Theorem 1 , and shall prove it by a direct method where it is needed . Remark 2 .

The theorem is false if no boundedness restriction is imposed on the coefficient ...

We shall see , however , that this fact is needed in the course of the

**proof**ofTheorem 1 , and shall prove it by a direct method where it is needed . Remark 2 .

The theorem is false if no boundedness restriction is imposed on the coefficient ...

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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 |

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