MA5296 Lecture 1 Completion and Uniform Continuity Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore Tel (65)

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

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

COMPLETION 4 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 ? denote the set of equivalence classes in C, define a dense embedding of S into, and a metric on Discussion 10. Define a ‘nice’ E.R. on C, let 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

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

NORMED VECTOR SPACES 6 Discussion 17. What is a normed vector space ? Discussion 18. How is it related to a metric 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) Discussion 19. What is a Banach / Hilbert Space ?

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

FOURIER TRANSFORM 8 We consider the measure spacewhere is Lebesque measure on M Definition The Fourier Transform on M is the set of Lebesque measurable subsets of R and 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

BROWNIAN MOTION 9 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 34. Explain the central limit theorem Discussion 31. Use the concept of a probability space to define the concept of a random variable R.V. Discussion 35. Develop and use a MATLAB program to simulate Brownian motion Discussion 32. Define expectation & variance of R.V. Discussion 33. Define independence of 2 R. variables

STOCHASTIC INTEGRALS 10 Discussion 37. If g in D and f : R  R is Brownian motion, use the Riemann-Stieltjes Integral to define Discussion 38 Show that if g and h are in D then Discussion 36 Show that the set D of step functions on R with compact support is dense in the random variable 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