168 PRINCIPLES OF MATHEMATICAL ANALYSIS 14. Let f be a continuous real function on R' with the following properties: Oss(1)S1, S(t + 2) - f(t) for every 1, and (0stst) S() – (ISIS1). Put D(t)- (x(t), y(1)), where x(1) - y(1) -E2-(3*2n). Prove that O is continuous and that maps I- [o, 1] onto the unit square 1 c R2, If fact, show that maps the Cantor set onto 1. Hint: Each (xo, yo) e 1 has the form Yo -22-"az. where each a, is 0 or 1. If to- 3-1-(2a,) show that f(3%) - a, and hence that x(to) - Xe, y(to) -m yo. (This simple example of a so-called "space-filling curve" is due to I. J. Schoenberg, Bull. A.M.S., vol. 44, 1938, pp. 519.) 15. Suppose fis a real continuous function on R', f.(t) -S(nt) for n=1, 2, 3, ..., and ) is equicontinuous on [0, 1]. What conclusion can you draw about f? 16. Suppose (fa) is an equicontinuous sequence of functions on a compact set K, and Sn} converges pointwise on K. Prove that (Sa) converges uniformly on K. 17. Define the notions of uniform convergence and equicontinuity for mappings into any metric space. Show that Theorems 7.9 and 7.12 are valid for mappings into any metric space, that Theorems 7.8 and 7.11 are valid for mappings into any complete metric space, and that Theorems 7.10, 7.16, 7.17, 7.24, and 7.25 hold for vector-valued functions, that is, for mappings into any R. 18. Let (S) be a uniformly bounded sequence of functions which are Riemann-inte- grable on [a, b), and put F.(x) m fa(t) dt (aSxsb). Prove that there exists a subsequence (F) which converges uniformly on [a, b). 19. Let K be a compact metric space, let S be a subset of 8(K). Prove that S is compact (with respect to the metric defined in Section 7.14) if and only if S is uniformly closed, pointwise bounded, and equicontinuous. (If S is not equicontinuous, then S contains a sequence which has no equicontinuous subsequence, hence has no subsequence that converges uniformly on K.) let SEQUENCES AND SERIES OF FUNCTIONS 169 20. If S is continuous on (0, 1] and if | S(x)x" dx =0 (n- 0, 1, 2, ...), prove that f(x) =0 on (0, 1]. Hint: The integral of the product of f with any polynomial is zero. Use the Weierstrass theorem to show that r(x) dx = 0. 21. Let K be the unit circle in the complex plane (i.e., the set of all z with z= 1), and let A be the algebra of all functions of the form S(e") - Gneln (0 real). Then A separates points on K and vanishes at no point of K, but nevertheless there are continuous functions on K which are not in the uniform closure of . Hint: For every fe A " Colfelf di0