By Constantin Corduneanu

ISBN-10: 0387098186

ISBN-13: 9780387098180

ISBN-10: 0387098194

ISBN-13: 9780387098197

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**Extra info for Diuretika: Diuretika-Symposion Dusseldorf 1979**

**Example text**

47) proves that f (t) can be uniformly approximated on R by trigonometric polynomials. Therefore, AP1 (R, C) ⊂ AP(R, C). 40). Then we must show the completeness. 48). Let {f j (t); j ≥ 1} ⊂ AP1 (R, C) be a Cauchy sequence. 46). 50) k=1 with each series on the right-hand side absolutely convergent: ∞ |ajk | < ∞, j ≥ 1. 51) k=1 The assumption that λk ’s are the same for each f j is not a restriction. For all f j , j ≥ 1, we have a countable set of λk ’s. Their union is also countable. 50). The Cauchy property means fj − fm 1 < ε, j, m ≥ N (ε).

In other words, the restrictions of the functions of {xm } to (a, b) form a Cauchy sequence in L1 ((a, b), C). 78) on each ﬁnite interval (a, b) ⊂ R. The limit function x(t) is thus deﬁned on R, and it is locally integrable. 78) belongs to the space M . But x = (x − xm ) + xm with xm ∈ M . 76), one obtains |xm − x|M ≤ ε for m ≥ N (ε). This means that x − xm ∈ M. Therefore, x ∈ M , which ends the proof of the assertion that M endowed with the norm | · |M is a Banach space. The space M is a fairly large function space, and it does contain as subspaces all the spaces Lp (R, C), 1 ≤ p ≤ ∞.

The Marcinkiewicz function space M2 = M/L is a Banach space with the norm deﬁned by x + L M = x M , x ∈ M. 20. 2, the norm x + L M = inf{ y M ; y ∈ x + L} was deﬁned. 19. Indeed, taking into account that L = {x; x ∈ M, x M = 0}, we have, for any y ∈ x + L, y M ≤ x M + x0 M with x0 M = 0 because x0 ∈ L. Hence, y M ≤ x M . But y = x + x0 gives x = y − x0 , x0 ∈ L, which means x M ≤ y M . Therefore, x + L M = inf{ y M ; y ∈ x + L} = x M , and we see that all the elements in M that deﬁne an element in M2 = M/L have the same M-seminorm.

### Diuretika: Diuretika-Symposion Dusseldorf 1979 by Constantin Corduneanu

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