Download e-book for iPad: A short course on Banach space theory by N. L. Carothers

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By N. L. Carothers

ISBN-10: 0521842832

ISBN-13: 9780521842839

This brief direction on classical Banach house thought is a typical follow-up to a primary path on useful research. the themes lined have confirmed valuable in lots of modern learn arenas, resembling harmonic research, the speculation of frames and wavelets, sign processing, economics, and physics. The e-book is meant to be used in a complicated subject matters path or seminar, or for self sustaining examine. It deals a extra uncomplicated creation than are available within the current literature and comprises references to expository articles and recommendations for additional examining.

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1. If (xn ) is a basis for a Banach space X , then every Pn (and hence also every xn∗ ) is continuous. Moreover, K = supn Pn < ∞. Proof. Banach’s ingenious idea is to define a new norm on X by setting |||x||| = sup Pn x . n → x, it’s clear that |||x||| < ∞ for any x ∈ X . The rest of the details Since Pn x − required to show that ||| · ||| is a norm are more or less immediate. To show that the Pn are uniformly bounded, we want to show that |||x||| ≤ K x for some constant K (and all x ∈ X ). To this end we appeal to a corollary of the Open Mapping theorem: Notice that the formal identity i : (X, ||| · |||) − → (X, · ) is continuous since x = limn − →∞ Pn x ≤ |||x|||.

If S is “small enough,” that is, if T is “close enough” to I , then T should be an isomorphism on [xn ]. That this is true is a useful fact in its own right and is well worth including. 6. If a linear map S : X − → X on a Banach space X has S < 1, then I − S has a bounded inverse and (I − S)−1 ≤ (1 − S )−1 . Notes and Remarks 41 Proof. The geometric series I + S + S 2 + S 3 + · · · converges in operator norm to a bounded operator U with U ≤ (1 − S )−1 . That U = (I − S)−1 follows by simply checking that (I − S)U x = x = U (I − S)x for any x ∈ X.

This allows us to use the much simpler functions χ I in place of the Haar functions in certain arguments. For example, it’s now very easy to see that The Haar System 31 the h n have dense linear span in L p [0, 1] for any 1 ≤ p < ∞. Also, please note that the set {χ I : I ∈ Ak } is again orthogonal in L 2 [0, 1] (although not the full set of χ I for all dyadic intervals I ). In particular, if Pk is the orthogonal projection onto span{h 0 , . . , h 2k+1 −1 }, and if Q k is the orthogonal projection onto span{χ I : I ∈ Ak }, then we must have Pk f = Q k f .

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A short course on Banach space theory by N. L. Carothers


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