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

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Nicholas Lawrance | ICRA 2011 1 Nicholas Lawrance | Thesis Defence 1 1 Functional Analysis I Presented by Nick Lawrance
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Transcript of Nicholas Lawrance | ICRA 20111Nicholas Lawrance | Thesis Defence1 1 Functional Analysis I Presented...

Nicholas Lawrance | ICRA 2011 1Nicholas Lawrance | Thesis Defence 11

Functional Analysis I

Presented by Nick Lawrance

Nicholas Lawrance | ICRA 2011 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

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Revision of topics/definitions

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Injective transformations

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Surjective transformations

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Bijective transformations

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Sequences

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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

1 : 1, 2,3,...X n n

{ 1,0,1}

I

X I

x

R; ; ;

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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

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

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lp - norms

• For an n-dimensional space

• 2-dimensional Euclidean space unit spheres for a range of p values

1 2

1 2

1

1

, , ,

, , ,

( , )

n

n

n pp

j jj

x

y

d x y

1

1 1 2 2( , ) 1p p pd x y

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-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

dimension 1

dim

ensi

on 2

lp norm based unit circles for 2-dimensional space

p = 0p = 0.5

p = 1

p = 2

p = 5

p = 20p = 1000000

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Metric Space

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Examples

• Euclidean R, R2, R3, Rn.• 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

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• Function space C[a,b]

• X is the set of continuous functions of independent variable t є J, J = [a,b]

tx

y

d(x, y)

ba

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lp-space

• Note that this basically implies that each point is a finite distance from the ‘origin’

• Sequence can be finite or not

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Open and closed sets

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• 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

Nicholas Lawrance | ICRA 2011 18

Open and closed sets

ε > 0

ε > 0

x0

x

xx0

B(x0, ε)

B(x0, ε)

Neighbourhood

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Selected problems

x0 x0+1x0-1 R

x0

C

tx0

ba

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0 1 2 3 4 5 6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t

x(t)

y(t)

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• We need

• Let f(t) = |x(t) – y(t)|• Find the stationary points

[0,2 ]

max ,t

d x t y t

sin cos

cos sin 0

cos sin [0,2 ]

3 7,

4 4

2

f t t t

dft t

dtt t t

t

f t

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0 1 2 3 4 5 60

0.5

1

1.5

t

|sin

(t)

- co

s(t)

|

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0 1 2 3 4 5 6-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

B(x, 2)

xy

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Accumulation points and closure

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Accumulation points and closure

B(x0, ε)

M X

x0

• Accumulation point if every neighbourhood of x0 contains a y є M distinct from x0

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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}

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Convergence

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Completeness

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Isometric mapping

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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))