MA5296 Lecture 1 Completion and Uniform Continuity

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MA5296 Lecture 1Completion and Uniform Continuity

• Wayne M. Lawton
• Department of Mathematics
• National University of Singapore
• 2 Science Drive 2
• Singapore 117543

Email matwml_at_nus.edu.sg http//www.math.nus/matwm
l Tel [phone]
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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
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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.
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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
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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
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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)
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FUNCTION SPACES

Definition A measure space is a triplet
Discussion 21. What are its three elements ?
measurable ?
Discussion 22. When is
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.
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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
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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
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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
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