New PDF release: Ideal Spaces

By Martin Väth (auth.)

Ideal areas are a really basic category of normed areas of measurable features, together with e.g. Lebesgue and Orlicz areas. Their most vital software is in useful research within the thought of (usual and partial) vital and integro-differential equations. The booklet is a slightly entire and self-contained advent into the overall conception of perfect areas. a few emphasis is wear areas of vector-valued features and at the optimistic point of view of the idea (without the axiom of choice). The reader must have easy wisdom in useful research and degree theory.

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Assume, X is semi-perfect, but not almost c~-perfect for any c~ > 0. 7 for any k there exists a sequence of nonnegative functions xnk E X , which converges monotonically increasing to x k E X with Ilxkll > k 3 lira llx ll = k3ck. By x k r 0 we have Ck > 0. Since xnk is monotonically increasing we have Ilxknll <- Ck. Thus, the sequence Yn = ~-]~2 1 k-2cklXkn is monotonically increasing and has bounded norm: z . 7 now implies y = supyn E X, but on the other hand we have by y >_ k - 2 c ~ l x k that ilyll > k-2cklk3Ck = k for any k, which is not possible.

The following corollary is due to Zabrejko [50, Theorem 3 and L e m m a 2]. 2 are simpler (although they follow the same line). 1. Each pre-ideal space with finite support has the W-property. 1 The W-Property 31 T h e corollary is s o m e w h a t surprising, if you observe t h a t if X is a pre-ideal space over some m e a s u r e space (S, ~ , #), then X depends only on ~ , not on any values of # (except for the null sets). Thus you m a y r e n o r m # a r b i t r a r y (without changing the null sets), and still have the same pre-ideal space.

X3 = (1, 1, 1,0, 0 , . . ) , . . e. to x = ( 1 , 1 , . . ) , but x • X . In other words: You can not prove 56 3. Ideal Spaces with Additional Properties the monotone convergence theorem by Vitali's theorem and Lebesgue's dominated convergence theorem without using special properties of X. 17] needs some completion (indeed, the proof uses the fact that the estimate L fn(s)#(ds) ~_ ~ fN(S)It(ds) + e implies lim sup f fn(s)#(ds) < lira sup f tt(E)---~O J E fN(s)#(ds)+ c uniformly in n, #(E)-+O J E which is not true in general without other assumptions).

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