Nicholas Lawrance | Thesis Defence

1. Functional Analysis I Presented by Nick Lawrance Nicholas Lawrance | Thesis Defence 1 1

2. What we want to take from this... • My hope is that a proper understanding of the fundamentals will provide a good basis for future work • Clearly, not all of the maths will be directly useful. We should try to focus on areas that seem like they might provide utility • The topic areas are not fixed yet

3. Revision of topics/definitions

4. Injective transformations

5. Surjective transformations

6. Bijective transformations

7. Sequences

8. Sequences • N = {1, 2, 3, ...} is countably infinite • The rational numbers Q are countable, the real numbers R are not • Examples • Can also have a finite index set, and a subset of the index results in a family of elements

9. Supremum and Infimum • Easy to think of as maximum and minimum, but not strictly correct. They are the bounds but do not have to exist in the set A = {-1, 0, 1} sup(A) = 1 inf(A) = -1 B = {n-1: n = [1, 2, 3, ...]} sup(B) = 1 inf(B) = 0

10. lp- norms • For an n-dimensional space • 2-dimensional Euclidean space unit spheres for a range of p values

12. Metric Space 12

13. such that Examples • Euclidean R, R2, R3, Rn. • Complex plane C • Sequence space l∞ • Remember a sequence is an orderedlist of elements where each element can be associated with the natural numbers N • Discrete metric space

14. Function space C[a,b] • X is the set of continuous functions of independent variable tєJ, J = [a,b] y x t a b d(x, y)

15. lp-space • Note that this basically implies that each point is a finite distance from the ‘origin’ • Sequence can be finite or not

16. Open and closed sets

17. Balls cannot be empty (they must contain the centre which is a member of X) • In a discrete metric space, sphere of radius 1 contains all members except x0, S(x0, 1) = X- x0

18. Open and closed sets ε > 0 x x0 B(x0, ε) ε > 0 Neighbourhood x x0 B(x0, ε)

20. x0 t a b Selected problems x0-1 x0 x0+1 R C x0

22. We need • Let f(t) = |x(t) – y(t)| • Find the stationary points

25. Accumulation points and closure

26. Accumulation points and closure • Accumulation point if every neighbourhood of x0 contains a y є M distinct from x0 B(x0, ε) x0 X M

27. Closure of the integers is the integers • Closure of Q is R • Closure of rational C is C • Closure of both disks is {z ||z|≤ 1}

28. Convergence

29. Completeness

30. Isometric mapping

31. Summary • Metric Spaces • Open closed sets (calls, spheres etc) • Convergence • Completeness • Next • Banach spaces (basically vector spaces) • Hilbert spaces (Banach spaces with inner product (dot product))