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

Nicholas Lawrance | ICRA 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

Nicholas Lawrance | ICRA Revision of topics/definitions

Nicholas Lawrance | ICRA Injective transformations

Nicholas Lawrance | ICRA Surjective transformations

Nicholas Lawrance | ICRA Bijective transformations

Nicholas Lawrance | ICRA Sequences

Nicholas Lawrance | ICRA 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

Nicholas Lawrance | ICRA 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

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

Nicholas Lawrance | ICRA

Nicholas Lawrance | ICRA Metric Space

Nicholas Lawrance | ICRA 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

Nicholas Lawrance | ICRA 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

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

Nicholas Lawrance | ICRA Open and closed sets

Nicholas Lawrance | ICRA 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

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

Nicholas Lawrance | ICRA

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

Nicholas Lawrance | ICRA

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

Nicholas Lawrance | ICRA

Nicholas Lawrance | ICRA

Nicholas Lawrance | ICRA Accumulation points and closure

Nicholas Lawrance | ICRA 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

Nicholas Lawrance | ICRA 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}

Nicholas Lawrance | ICRA Convergence

Nicholas Lawrance | ICRA Completeness

Nicholas Lawrance | ICRA Isometric mapping

Nicholas Lawrance | ICRA 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))