## Linear Operators: General theory |

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Page 132

It follows easily from Definition 12 that if a , aina n = 1 , 2 , ... , are finite countably

additive measures with an ( E ) + 2 ( E ) , Ee , and if an , n = 1 , 2 , ... are u -

Let ...

It follows easily from Definition 12 that if a , aina n = 1 , 2 , ... , are finite countably

additive measures with an ( E ) + 2 ( E ) , Ee , and if an , n = 1 , 2 , ... are u -

**singular**, then 2 is also u -**singular**. 14 THEOREM . ( Lebesgue decomposition )Let ...

Page 564

Show that a set of solution vectors yı ( t ) , . . . , Yn ( t ) of dy / dt = A ( t ) y form a

basis for the solution space if and only if they form the columns of a solution

matrix Y ( t ) of dY / dt = A ( t ) Y which corresponds to a non -

matrix Y . .

Show that a set of solution vectors yı ( t ) , . . . , Yn ( t ) of dy / dt = A ( t ) y form a

basis for the solution space if and only if they form the columns of a solution

matrix Y ( t ) of dY / dt = A ( t ) Y which corresponds to a non -

**singular**initialmatrix Y . .

Page 855

3 ( 167 ) , III . 9 . 46 ( 174 ) Simple Jordan curve , ( 225 )

ring , ( 40 ) non -

12 ( 131 ) derivatives of , III . 12 . 6 ( 214 ) Lebesgue decomposition theorem , III .

3 ( 167 ) , III . 9 . 46 ( 174 ) Simple Jordan curve , ( 225 )

**Singular**element in aring , ( 40 ) non -

**singular**operator , ( 45 )**Singular**set function , definition , III . 4 .12 ( 131 ) derivatives of , III . 12 . 6 ( 214 ) Lebesgue decomposition theorem , III .

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero