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By J.H. Jeans, M.A., F.R.S

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Clearly A A = = [S]. We will denote the set A by (x,y). If x = c, yy = = d, with c, dd G £ IR, there is now an ambiguity of notation so we adopt the convention that (c, d) is the real interval in this case, and the internal interval is denoted by *(c, d). ) (4) Let z gG *IR. *K. Then the function 7 : *IR -> *IR given by 7(3) = "sin(z3) is internal. If z = = [a], x = = [bj, [b], then 7(3) 7(1) = [sin(a(-)6(-))]. In other words, 7 = [g] with g(i) being the function r i-» sin(a(i)r). Interesting cases are where z is infinitesimal or infinite.

2 (b). oo. 6 Let G £ *L{H ) ) with | G | HH KK , H N < oo. HAT - Proof For For O H , \GPTN"V\ \V\ < oo, oo, so GPTN*V » eE H |GPrw"t>| < | G | H WN , H WN M GPiN*v is weakly nearstandard in H and we may define an operator g by gv = °(GPTN*V), °(GPr^*t)), which is clearly linear. | so p GL ( H , H ) and | 5 | HH,,H H < °°|G| | G | H „,H N , H K. 2 (a), since gv {u,gv) {u,gv) so that g = °G as required. so that g = °G as required. «w «w w w GPr^'v in H, we have that for any u G H GPr^'v in H, we have that for any u £ H (*u,GPTN*V) (*u,GPTN*V) © © A Crucial Lemma The following crucial lemma has many applications.

K=l The space HAT plays a crucial role because it is hyperfmite dimensional, so by the Transfer Principle all finite dimensional techniques can be used, and at the same time HAT is rich enough to represent H (and other related spaces) by means of standard part mappings discussed below. By PrAf Prjv we denote the orthogonal projection of *H onto HATHAT. We use U, V to denote elements of HAT and we write Uk — (U, Ek) for k £ *1N. Then |t/|2 = f> 2 , N (U,V) = Y/UkVk, fart k=l *=1 k=l N N ((U, UkV ((U,V)) V))==£ Y,XkW k, k, kV 2 2 \\U\\ = £= £ \kUl \\U\\ \kUl k=l k=\ an where (■,■), (-,-), | • |, ((■,•))> d II ' II> II, denote the extensions of the standard operations to *H, and (^k)ke'TN is the extension of the sequence (Afc)fce]N.

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