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If µ ∈ Sloc, m(µ) < ∞, and µ satisfies (14) for t = t0,t0 1 with some t0 > 0, then µ ∈ S2 loc The organization of this paper is as follows In Sect 2, we give preliminaries for the proof of the results In Sect 3, we prove Theorem 11 and its corollary. Real Analysis Homework #1 Yingwei Wang ∗ Department of Mathematics, Purdue University, West Lafayette, IN, USA 1 Banach space Question Let (xn) ⊂ X be a Banach space, and P∞ n=1 kxnk is convergentProof that. ý ¥ " Ð µ â 1 ¥ â # e p Û e p Ý.

C Z k ` O } K d \ , q U d } , Q z þ S , c Z k ` O } ¸ c E E ä ® ê È ?. CDE B A ?@ K J I H G F& L MNOP , Q R S TU VWX YZ \ YZ ^ _ _ _ ` _ a b c dd f ea _ g _ _ h d i g _ _ _ a b c dd f ea. Classiflcation of C(M)£µ Zp for certain pairs (M;µ) 133 induction, we can flnally flnd U0 2 U0(Mp(C(X))) and v 2 U(C(X)) such that U = U0 diag(1p¡1;v) According to 21, fx 2 Mjµk(x) = xg = Mµ, k = 2;¢¢¢ ;p ¡ 1 Now set M0 = MnMµLet M=µ (resp M0=µ)) denote the orbit space of µ and let P be the canonical projective map of M (or M0) onto M=µ (or M0=µ)Let Oµ(x) = fx;µ(x.

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