MA5296 Lecture 1 Completion and Uniform Continuity

1. MA5296 Lecture 1Completion and Uniform Continuity Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg http://www.math.nus/~matwml Tel (65) 6516-2749 1

2. EXTENSION OF THE NUMBER CONCEPT denote set of natural numbers (axiomatically described by Giuseppe Peano in 1889), ring of integers, and rational / real / complex fields. Discussion 1. Cartesian Product of Sets, Ordered Pairs Discussion 2. Equivalence Relations on Sets Discussion 3. Construct Z from N using equivalence classes of pairs (m,n) in N x N, Q from Z, C from R Discussion 4. Construct R from Q using equivalance classes of Cauchy Sequences in Q 2

3. METRIC SPACES defined by Maurice Frechet in 1906, are pairs (S,d), where S is a set and d a distance function that satisfies: Discussion 5. What three properties ? Discussion 6. Show that (R,d(x,y)=|x-y|) is a M. S. Discussion 7. What is a topological space ? Discussion 8. Show that every M. S. is a T. S. 3

4. COMPLETION Let (S,d) be a metric space and C denote the set of Cauchy Sequences f : N  S Discussion 9. Explain what property f must have ? Discussion 10. Define a ‘nice’ E.R. on C, let denote the set of equivalence classes in C, define a dense embedding of S into , and a metric on Definition A M.S. is complete if every C.S. converges Discussion 11. Prove that the construction in 10 gives a complete metric space Discussion 12. For every prime p in N, explain how to construct the p-adic completion of Q 4

5. UNIFORM CONTINUITY Let (S,d) and (X,p) be metric spaces and f : S  X Discussion 13. When is f uniformly continuous ? Discussion 14. Show that f U.C.  f maps CS to CS Discussion 15. Prove that f U.C.  f satisfies the Extension Principle there exists that is U.C. and the following diagram commutes Discussion 16. Use the E.P. to define 5

6. NORMED VECTOR SPACES Discussion 17. What is a normed vector space ? Discussion 18. How is it related to a metric space ? Discussion 19. What is a Banach / Hilbert Space ? Definition If X is a compact topological space C(X) denotes the set of complex valued continuous functions on X. Discussion 20. Construct a Banach Space on C(X) 6

7. FUNCTION SPACES Definition A measure space is a triplet Discussion 21. What are its three elements ? Discussion 22. When is measurable ? Discussion 23. Define an E.R. on the set of such f Discussion 24. Define a vector space on the set of E.C. Definition For define the set of E.C. Discussion 25. Construct a Banach Space on this set. 7

8. FOURIER TRANSFORM We consider the measure space where M is the set of Lebesque measurable subsets of R and is Lebesque measure on M Definition The Fourier Transform on is the function Discussion 26. Show that T([f]) in C(R) Discussion 27. Show that T([f]) depends only on [f] Discussion 28. Show that T([f])(y) 0 as |y| increases Discussion 29. Show density of Discussion 30. Show that the F.T. is an isometry then use the E.P. to extend it to a map 8

9. BROWNIAN MOTION Discussion 31. Use the concept of a probability space to define the concept of a random variable R.V. Discussion 32. Define expectation & variance of R.V. Discussion 33. Define independence of 2 R. variables Discussion 34. Explain the central limit theorem Definition A Brownian motion is a random process f : R  R such that (1) for every interval I = [a,b] the random variable f(b) – f(a) (called the jump over I) is Gaussian with mean 0 and variance b-a, and (2) the jumps of f over disjoint intervals are independent Discussion 35. Develop and use a MATLAB program to simulate Brownian motion 9

10. STOCHASTIC INTEGRALS Discussion 36 Show that the set D of step functions on R with compact support is dense in Discussion 37. If g in D and f : R  R is Brownian motion, use the Riemann-Stieltjes Integral to define the random variable Discussion 38 Show that if g and h are in D then Discussion 39 Use the E. P. to define I(g) for Discussion 40 Define the Ito Integral and explain how it extends the stochastic integral I defined above 10