## Linear Operators: Spectral theory |

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Page 1179

sends a scalar - valued function with the Fourier transform f ( ) into the vector -

valued function whose nth component has the Fourier transform In ( 5 ) defined

by ...

**Proof**. We saw in the course of proving Theorem 25 that the mapping MK whichsends a scalar - valued function with the Fourier transform f ( ) into the vector -

valued function whose nth component has the Fourier transform In ( 5 ) defined

by ...

Page 1724

) = ( 1 , 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 f and g are in CO ( I ) . It follows ...

**Proof**. By the preceding lemma and by Corollary 11 it suffices to show that ( Tj , g) = ( 1 , 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 f 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 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

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