## Linear Operators: Spectral theory |

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Page 1660

Of course , if we deal only with

identifications are made and the phrases “ open set , " " closed ... If I is an open

. .

Of course , if we deal only with

**subsets**of the interior of the cube C in E " , noidentifications are made and the phrases “ open set , " " closed ... If I is an open

**subset**of C , in the modified sense explained in the preceding paragraph , then C. .

Page 1663

for each open

DEFINITION . Let I be an open

set of all F in D ( 1 ) for which ( F ( 0 ) \ Fl ( - k ) = sup " < oo PEC , o ( I ) 19luk ) will

...

for each open

**subset**1 , of I whose closure is compact and contained in I . 35DEFINITION . Let I be an open

**subset**of C and let k be a positive integer . ( i ) Theset of all F in D ( 1 ) for which ( F ( 0 ) \ Fl ( - k ) = sup " < oo PEC , o ( I ) 19luk ) will

...

Page 1695

M Lemma 13 and the following lemma taken together give considerable insight

into the nature of distributions in general . 15 LEMMA . Let F be a distribution in

the open

union ...

M Lemma 13 and the following lemma taken together give considerable insight

into the nature of distributions in general . 15 LEMMA . Let F be a distribution in

the open

**subset**I of E " . Let { In } be a sequence of open**subsets**of I whoseunion ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

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additive adjoint operator Akad 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 determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f 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 Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero