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By Solomon Lefschetz

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As n can be made indefinitely large, the range of the embedding cannot lie in any Euclidean space. D In the same setting it is clear that the range of an embedding in Hilbert space is a bounded set if and only if the function F is bounded. The following result provides conditions which insure the boundedness of a function in N(IR). 4. 27) Suppose F is bounded. 24) that F2(r) ~ r 2a({0}) for r > 0; hence a({O}) = O. Next let Proof F,jt) = J,r a sin 2tu ~ da(u) s: t~ r _Ja (! _ (0 < e < a < (0). The mean-value of F"a over the interval [O,T] may be written as ~ I: F•.

1-~ II Choose~,O<~~I,sothat 1+(r_l)~=Al/Ao. oY) ~. 54 Chapter III. lyll =(1 +(r-l)oc)r. Hence 1+(r-l)oc::;; 1, or r::;; 1. 0 and (b) holds. In this situation, set Xn = x and Yn = (+'YnY 11 . 2' Yn-Xn = X+'YnY- X = 'YnY. ylI) (). ~ 0). ) = 2. n) = 'Yn for all large n. nyll)ylI = 1 (n ~ 1). n. IIX-Anyll Ilx-),nYll Furthermore, all large n. 7. 1). If Xt>X2EX and Yl,y2EY such that IIxdl = IIX211 = IIYlll = IIhll = 1, XlNx2 and y l NY2, then IlaYl + bY2 II ::;; Ilaxl +bX211 for all real a, b. Moreover normality is symmetric in both X and Y.

L) (0 < p ~ 2) has infinite (linear) dimension. l) (0 < p ~ 2) if and only if there exists a finite positive measure ex on \R+ such that F(r) = f~ e- rPu dex(u) (r ~ 0). l-measure. To do this suppose that such sets E 1 ,E2 , ••• ,Ek have been constructed. l measure. The existence of the sequence now follows by induction. l» and let ej be the characteristic function of Ej> j = 1,2, .... 12) the function rllx-yll~u (u> 0) is positive definite on U for 0 < p ~ 2. l). For the converse, suppose Fe RPD(U).

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