Is a Metric on the Reals R

Definitions: Let S be a set, f : S  R be a function, and ( f n ) be a sequence of functions from S into R.

(a) converges pointwise on S to f iff, for each x in S, f n (x )  f (x ) in R. Equivalently,

 x  S  ε  0  m(x)  N, n  m(x)  f n (x)  f (x)  ε .

(b) converges uniformly on S to f iff  ε  0  m  N  x  S, n  m  f n (x)  f (x)  ε .

Note: (i) If converges uniformly on S to f then converges pointwise on S to f.

n (ii) Pointwise convergence need not imply uniform convergence. For example f n (x )  x converges 0 if 0  x  1 pointwise on [0,1] to f (x )   , but convergence is not uniform. 1 if x  1

Definitions: Let S be a set. (a) A function f : S  R is (uniformly) bounded iff there is a number b so that f (x )  b for all x in S. (b) The uniform norm of a bounded function is f  min { b :x  S f (x )  b }  sup { f (x ) : x  S} . u (c) B(S) denotes the set of all bounded functions .

Theorem: (a) B(S) is a vector space under the pointwise operations (f  g)( x )  f (x )  g(x ) and ( cf )( x )  c f (x ). (b) The uniform norm is a norm on B(S). (c) converges to f in the metric space B(S) iff converges uniformly on S to f. (d) B(S) is a complete under the uniform norm. A complete normed space is called a Banach space.

Theorem: If f : S  T is continuous and K  S is compact, then f (K )  {f (x ): x  K } is a compact subset of T.

Corollary: If is continuous and S is compact then there are points p and q in S so that f (q)  f (x )  f (p) for all x in S. In particular f is bounded on S.

Definition and Theorem: For S a compact metric space let C(S) denote the set of all continuous functions . (a) C(S) is a vector space under the pointwise operations (b) If is a sequence in C(S) and converges uniformly on S to f, then f is continuous. (c) C(S) is a closed subset of B(S) and thus is complete under the uniform norm.

2 2 Lemma: If g is continuously differentiable on [a,b] then g  2 g g  (b  a ) 1 g . u 2 2 2

Theorem: Let be a sequence of continuously differentiable functions on [a,b]. If (i) is Cauchy in 2-norm, and (ii) the derivatives ( f n ) are bounded in 2-norm, then ( f n ) converges uniformly to a continuous function on [a,b].

Theorem: Let be a sequence of continuously differentiable functions on [a,b]. If

(i) lim f n (x 0 ) exists for some x 0 in [a,b], and (ii) converges in 2-norm to some continuous g, then there is a differentiable f so that f   g and converges uniformly to f on [a,b].

PROBLEMS

Problem 7-1: Answer these questions for each sequence . Does the sequence converge pointwise on S? If so is the pointwise limit continuous? Is convergence uniform?

1 n 1 (a) f n (x )  , S  [0,1] (b) f n (x )  , S  [0,1] (c) f n (x )  tan (n x ), S  R 1  n x n  x

Problem 7-2: In B[0,1] let f n (x )  max {1  n x , 0 }.

(a) Calculate f n u and f n  f m u for n  m .

(b) Is the set M  { f 1 , f 2 , f 3 , ... , f n , ... } totally bounded in B[0,1]? (c) Does the sequence converge in B[0,1]? If so find its limit.

Definition: A function F:S  T between metric spaces is an isometry iff d(F(x), F(z))  d(x, z) for all x and z in S. Note this doesn’t require that F is onto.

Problem 7-3: For (S,d) a metric space and x 0 a distinguished point in S, define F:S  B(S) by

F (x)(s)  d(x 0 , s)  d(s, x) Prove that F is an isometry with values in C(S).

Problem 7-4: Let be a sequence in C[a,b]. (a) Prove that if f n  0 uniformly then f n  0 in 2- norm. Give (or review) an example showing the converse is false. b b (b) Prove that if in 2-norm then f n (u) d u  0 and f n (u) d u  0 . Give (or review) an a a example showing the converse is false.

Problem 7-5: If g is continuously differentiable on [a,b] and c  1  1/(b  a) then

2 2 g  c { g  g }1/2 . u 2 2

2 2 Problem 7-6: On [1, 1] let f n (x)  f( n x) for f(x)  x sgn (x) /(1  x ) .

2 2 d u (a) f  sgn  n  0 . n  0 2 2 2 n (1  u ) (b) ( f n ) can’t converge in 2-norm to a continuous function on [1, 1] .

2 u 2 d u (d) f   8 n n   . n  0 2 4 2 (1  u )

2 Problem 7-7: On let f n (x)  x  (1/n) . (a) converges uniformly to x on . (b) converges to sgn(x) pointwise and in 2-norm on .

Definition (a Cantor Set): There is a recursively defined sequence (C n ) of subsets of [0,1] with the following properties: (i) C 0  [0,1] ; (ii) C n is the union of a finite number of disjoint closed bounded intervals, called the components of C n , and; (iii) C n 1 is obtained from by removing the open middle third of each component of . For instance C1  [0, 1/3]  [2/3, 1] ,  C1  [0, 1/9]  [2/9, 1/3]  [2/3, 7/9]  [8/9, 1] , and so on. The Middle Third Cantor Set is C   n 1 C n .

n  k Problem 7-8: Write S  { 0,1}N and define f :S  R by f (x)  2 3 x(k) . n n  k 1 (a) Show that converges uniformly and in C(S) to a Lipschitz function f :S  R . (b) Show that f is one-to-one and maps onto the Middle Third Cantor Set C. (b) Show that the inverse function f 1 :C  S is continuous. A continuous, one-to-one and onto function with continuous inverse is called a homeomorphism.