Y is a linear and closed II Txll and we remark that property (1) in Zabreiko's lemma is obvious. For the second property, suppose that (x n) has the property that L F(x n) converges (in the opposite case there is nothing to prove). 1 n-+'x i= 1 and since T is a closed operator we have and thus which gives the second property of F. Thus F is continuous and this is clearly 0 equivalent with the continuity of T. We now give briefly some applications of these theorems.
3) a(M) = 0 iff M is compact. (4) If M ~ N then a(M) < a(N). Now we have the following characterization of completeness of a metric space using the Kuratowski measure of noncompactness. 2 (Kuratowski, 1929). A metric space (S, d) is complete ill the if (FII)r is a sequence of closed subsets ()f S with the following assertion holds: properties: (a) Fl~F2~"'~Fn~FII+1~"', (b) lim a(F II ) = 0 then n:~lFn is nonempty (and compact). For an interesting proof of this result, see Pasicki (1979). For other measures of noncompactness, as well as for their use in fixed point theory, the reader may consult the author's book on fixed point theory, Istratescu (1981).