By Gray A., Mathews G.B.

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**Extra info for A treatise on Bessel functions and their applications to physics**

**Sample text**

L is a measure of noncompactness. L(B). Let us now prove the above inequalities: Since Pn is a compact operator we have X(PnB) = for every bounded subset B of X. In view of RnB c B + (-PnB) and B c RnB + PnB, we obtain X(RnB) ::; X(B) and X(B) ::; X(RnB), and thus X(B) = X(RnB). Now the obvious inequality X(RnB) ::; IIRnBII for all n E N implies X(B) ::; infnENIiRnBII· Let us now assume that X(B) = X(RnB) = r and let e > 0. There exists a finite set Bo such that B c UXEBo B(x, r + e). Let x E Band Xo E Bo such that Ilx - xoll < r + e.

Ek, 0, ... J < 1. Therefore, we see that diam(A) ~ J2, and since ,B(A) ~ diam(A), we can conclude that ,B(A) ~ J2. In fact ,B(A) = J2. To see this, we note that the subsets Am of A given by Am = {u nm : n E N} have their elements equidistant and so m-l rn they are a-minimal sets. Hence for them ,B(Am) = a(Am) = - - v 2 and thus ,B(A) ~ sup{ a(Am) : mEN} = m J2. However, there is no ,B-minimal subset B of A such that ,B(B) = ,B(A).

L' f3(B) + c) contains an infinite subset A m+1 cAm n {Xn : n > n m }. +l E Am. The sequence {x nk } satisfies the condition d(xnj,x nk ) < f3(B) + c for every k, j EN. 4. The MNC f3 is algebraically semi-additive. 27 CHAPTER II. MEASURES OF NONCOMPACTNESS Proof. Let Bl and B2 be arbitrary bounded subsets of a Banach space X. Fix any e > 0 and let {zn} be a (f3(B l + B 2) - c)-separated sequence in Bl + B 2. We write Zn = Xn + Yn, where Xn E Bl and Yn E B 2. 3 we can extract a subsequence {x nk } of {xn} such that Ilxnk - xnj II < f3(Bd + e for every k, j E N.

### A treatise on Bessel functions and their applications to physics by Gray A., Mathews G.B.

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