1. Nicholas Lawrance | ICRA 20111Nicholas Lawrance | Thesis Defence1 1 Functional Analysis I Presented by Nick Lawrance

2. Nicholas Lawrance | ICRA 20112 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. Nicholas Lawrance | ICRA 20113 Revision of topics/definitions

4. Nicholas Lawrance | ICRA 20114 Injective transformations

5. Nicholas Lawrance | ICRA 20115 Surjective transformations

6. Nicholas Lawrance | ICRA 20116 Bijective transformations

7. Nicholas Lawrance | ICRA 20117 Sequences

8. Nicholas Lawrance | ICRA 20118 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. Nicholas Lawrance | ICRA 20119 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) = 1inf(A) = -1 B = {n -1 : n = [1, 2, 3,...]}sup(B) = 1inf(B) = 0

10. Nicholas Lawrance | ICRA 201110 l p - norms For an n-dimensional space 2-dimensional Euclidean space unit spheres for a range of p values

11. Nicholas Lawrance | ICRA 201111

12. Nicholas Lawrance | ICRA 201112 Metric Space

13. Nicholas Lawrance | ICRA 201113 Examples Euclidean R, R 2, R 3, R n. Complex plane C Sequence space l ∞ –Remember a sequence is an ordered list of elements where each element can be associated with the natural numbers N Discrete metric space such that

14. Nicholas Lawrance | ICRA 201114 Function space C[a,b] X is the set of continuous functions of independent variable t є J,J = [a,b] t x y d(x, y) ba

15. Nicholas Lawrance | ICRA 201115 l p -space Note that this basically implies that each point is a finite distance from the ‘origin’ Sequence can be finite or not

16. Nicholas Lawrance | ICRA 201116 Open and closed sets

17. Nicholas Lawrance | ICRA 201117 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 x 0, S(x 0, 1) = X- x 0

18. Nicholas Lawrance | ICRA 201118 Open and closed sets ε > 0 x0x0 x x x0x0 B(x 0, ε) Neighbourhood

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20. Nicholas Lawrance | ICRA 201120 Selected problems x0x0 x 0 +1x 0 -1 R x0x0 C t x0x0 ba

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22. Nicholas Lawrance | ICRA 201122 We need Let f(t) = |x(t) – y(t)| Find the stationary points

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25. Nicholas Lawrance | ICRA 201125 Accumulation points and closure

26. Nicholas Lawrance | ICRA 201126 Accumulation points and closure B(x 0, ε) M X x0x0 Accumulation point if every neighbourhood of x 0 contains a y є M distinct from x 0

27. Nicholas Lawrance | ICRA 201127 a)Closure of the integers is the integers b)Closure of Q is R c)Closure of rational C is C d)Closure of both disks is {z | |z| ≤ 1}

28. Nicholas Lawrance | ICRA 201128 Convergence

29. Nicholas Lawrance | ICRA 201129 Completeness

30. Nicholas Lawrance | ICRA 201130 Isometric mapping

31. Nicholas Lawrance | ICRA 201131 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))