INTRODUCTION

Week 1 Lecture 1.1

Grading

•  Each week: –  Video lectures –  PDF reading material –  Quizzes and peer-graded problems

•  Use the Forum!

•  There will be a final exam.

•  Videos are stand-alone: you can do this class without the PDF material, but your score will be capped at 80%.

•  Passing grade is 60%. This is what you need to get your certificate.

What we will learn this week:

•  What is the purpose of functional analysis? •  What is a topology? •  What is an open set and a closed set? •  What is a compact set? •  What is convergence?

Convergence?

un = 1/n u1 = 1 u2 = 1/2 u3 = 1/3 u4 = 1/4 u10 = 1/10

1 2 3 4 5 6 7 8 9 10

1

Convergence?

…

Convergence?

fn defined from R to R by

fn(x) = |sin(x)|n

What we will learn this week

•  What is the point of functional analysis? •  What is a topology? •  What is an open set, a closed set? •  What is a compact set? •  What is convergence? •  What is continuity? •  What is the initial topology? •  Trick your calculus instructor: every function can

be continuous!

WHAT IS THE PURPOSE OF FUNCTIONAL ANALYSIS?

Week 1 Lecture 1.2

OPEN SETS & TOPOLOGY

Week 1 Lecture 1.3

Definition: Topology

Let X be a set and T be a family of subsets of X. T is called a topology on X if: i.  The empty set Ø and X are elements of T ii.  Any union of elements of T is in T iii.  Any finite intersection of elements of T is in T

(X,T) is a topological space. Elements of T are called open sets.

In-video Quiz

X = { 1 , 2 , 3 , 4, 5 } T = { {1,2} , {3,4} } Is T a topology?

In-video Quiz

X = { 1, 2 , 3 , 4 , 5 } Which ones are topologies? T = { Ø , {1,2} , {3,4} , {1,2,3,4}, X } T = { Ø , {1,2} , {2,3} , {1,2,3,4}, X } T = { Ø , {1,2} , {2,3} , {2}, X } T = { Ø , {1,2} , {2,3} , {2}, {1,2,3}, X }

Let us answer this more general question How do you turn a family of sets F into a topology? (while adding the fewest possible sets)

①  Add Ø and the whole space to F. ②  Add to F all finite intersections of elements of F ③  Add to F all unions of elements of (new) F F is now stable by unions It can be proved F is stable by finite intersections.

Construction of a Topology

 

Let us answer this more general question How do you turn a family of sets F into a topology? (while adding the fewest possible sets)

①  Add Ø and the whole space to F. ②  Add to F all finite intersections of elements of F ③  Add to F all unions of elements of (new) F    

Construction of a Topology

Note: 2 and 3 cannot be permuted!

Definition: Topology

A set X is always a topological space… The trivial topology: T = {Ø,X} The discrete topology: T = {all subsets of X} Given two topologies on X: T1 and T2 with T1 ⊂ T2 . T1 is coarser (or weaker or smaller) than T2

T2 is finer (or stronger or larger) than T1

If T is a topology then Tt ⊂ T ⊂ Td

d

t

In-Video Quiz

X = { 1 , 2 , 3 , 4, 5 } T1 = { Ø , {1,2} , {1,2,3,4}, X } T2 = { Ø , {1,2} , {5} , {1,2,5} , {1,2,3,4} , X } Which one is true? T1 is coarser than T2

T2 is coarser than T1

None of the above

Definition: Usual Topology on R

X = R T = { sets O s.t.

for every x in O, there exists ε>0, ]x-ε,x+ε[ ⊂ O }

Examples: ]1,2[ is open [1;2[ is not

1 2 x

1 2 x=

Definition: Closed sets

Complements of open sets are called closed sets. Example: X = { 1 , 2 , 3 , 4 , 5 } T = { Ø , {1,2} , {2,3} , {2} , {1,2,3} , X } {1,2} is an open set {3,4,5} is a closed set because {3,4,5} = X \ {1,2}

Definition: Neighborhood

Let X be a topological space Let x ϵ X. The set U is called a neighborhood of x if There exists an open set V s.t. i.  x ϵ V ii.  V ⊂ U The set of neighborhoods of x is noted V(x).

COMPACT SETS

Week 1 Lecture 1.4

Definition: Compactness

Let X be a topological space. We say that K⊂X is a compact set if K is not empty and for any arbitrary open sets Ui ⊂ X (i ϵ I) whose union contains K, one can find a finite number of these open sets such that their union contains K.

Example: Compactness

Let X=R be equipped with the usual topology. K=]0;1] is not compact Ui = ]1/i, 2[ for (i ϵ N\{0}) The union contains K    

Let x be in K Then 0<x≤1 Let n=[1/x]+1 Then n>1/x Therefore x>1/n Therefore x is in Un Thus K is in the union of Ui for i in N\{0}

0 1 x

Un

1/n

Example: Compactness

Let X=R be equipped with the usual topology. K=]0;1] is not compact Ui = ]1/i, 2[ for (i ϵ N\{0}) The union contains K    

             

Example: Compactness

Let X=R be equipped with the usual topology. K=]0;1] is not compact Ui = ]1/i, 2[ for (i ϵ N\{0}) The union contains K But a finite number of these open sets is not enough to cover K

             

Example: Compactness

Let X=R be equipped with the usual topology. Is K=[0;+∞[ a compact set? Ui = ]-1,i[ for (i ϵ N\{0}) The union contains K But a finite number of these open sets is not enough to cover K

Example: Compactness

Let X=R be equipped with the usual topology. K=[0;1] is compact. Let S be a set of open covers of K. Let A be the set of x in [0;1] such that one can extract a finite subcover of S for [0,x] A is not empty (it contains 0) A is bounded by 1

A has a supremum M

Example: Compactness

Let X=R be equipped with the usual topology. K=[0;1] is compact. [0,M] can be covered by a finite subcover of S Suppose M<1 Let O be in S containing M O is open, thus there exists ε>0 s.t. [M,M+ε[⊂O So we can build a finite subcover of [0,M+ε/2]

Example: Compactness

Let X=R be equipped with the usual topology. K=[0;1] is compact. [0,M] can be covered by a finite subcover of S Suppose M<1 It leads to a contradiction, therefore it is wrong Thus M=1

CONVERGENCE & CONTINUITY

Week 1 Lecture 1.5

Definition: Converging Sequences

Let X be a topological space and (xn) be a sequence of elements of X. We say that (xn) converges to l if (xn) may converge to several elements of X

∀V∈V (l), ∃N ∈ N, n ≥N⇒ xn ∈ V

A topological space X is a Hausdorff space (or a T2 space or a separated space) if: Given two distinct points in X one can find two open disjoints sets, each containing a point In a Hausdorff space, the limit of a sequence is unique.

Hausdorff Spaces

Definition: Converging Sequences

Let X be topological a space and (xn) be a sequence of elements of X. We say that (xn) converges to l if (xn) may converge to several elements of X  

 

∀V∈V (l), ∃N ∈ N, n ≥N⇒ xn ∈ V

Definition: Converging Sequences

Let X be topological a space and (xn) be a sequence of elements of X. We say that (xn) converges to l if (xn) may converge to several elements of X If the topology on X is stronger (larger/finer),

it is “harder” for (xn) to converge.  

∀V∈V (l), ∃N ∈ N, n ≥N⇒ xn ∈ V

Definition: Converging Sequences

Let X be topological a space and (xn) be a sequence of elements of X. We say that (xn) converges to l if (xn) may converge to several elements of X If the topology on X is stronger (larger/finer),

it is “harder” for (xn) to converge. If X is equipped with the discrete topology,

only sequences that become constant converge.

∀V∈V (l), ∃N ∈ N, n ≥N⇒ xn ∈ V

Definition: Continuous Mappings

Let X and Y be topological spaces A mapping f : X→Y is continuous if the inverse image of an open set is an open set. If the topology on X is finer (larger/stronger),

it is “easier” for f to be continuous. If X is equipped with the discrete topology,

any mapping is continuous.

Continuous Mappings and Sequences

The chosen definitions “work well”: Proposition Let X and Y be two topological spaces Let f : X→Y is be a continuous mapping Let (xn) be a sequence in X converging to l Define yn = f(xn) Then (yn) converges to f(l) in Y

Continuous Mappings and Sequences

Proof Let W be any neighborhood of f(l). Then there exists an open set U such that f(l) ϵ U and U ⊂ W Since f is continuous, f-1(U) is open (thanks to the definition of continuity) Since f(l) ϵ U, we have l ϵ f-1(U) We always have f-1(U) ⊂ f-1(W)

f-1(W) is a neighborhood of l

Continuous Mappings and Sequences

Proof Let W be any neighborhood of f(l).            

f-1(W) is a neighborhood of l

Continuous Mappings and Sequences

Proof Let W be any neighborhood of f(l).   V = f-1(W) is a neighborhood of l      

Continuous Mappings and Sequences

Proof Let W be any neighborhood of f(l). Since (xn) converges to l: V = f-1(W) is a neighborhood of l      

VxNnN,N(l),V n∈⇒≥∈∃∈∀ V

Continuous Mappings and Sequences

Proof Let W be any neighborhood of f(l). Since (xn) converges to l: V = f-1(W) is a neighborhood of l Thus, there exists N ϵ N s.t. n ≥ N implies xnϵ f-1(W)    

VxNnN,N(l),V n∈⇒≥∈∃∈∀ V

Continuous Mappings and Sequences

Proof Let W be any neighborhood of f(l). Since (xn) converges to l: V = f-1(W) is a neighborhood of l Thus, there exists N ϵ N s.t. n ≥ N implies xnϵ f-1(W) n ≥ N implies yn = f(xn) ϵ W. Hence:  

VxNnN,N(l),V n∈⇒≥∈∃∈∀ V

Continuous Mappings and Sequences

Proof Let W be any neighborhood of f(l). Since (xn) converges to l: V = f-1(W) is a neighborhood of l Thus, there exists N ϵ N s.t. n ≥ N implies xnϵ f-1(W) n ≥ N implies yn = f(xn) ϵ W. Hence:  

VxNnN,N(l),V n∈⇒≥∈∃∈∀ V

WyNnN,N(f(l)),W n ∈⇒≥∈∃∈∀ V

Continuous Mappings and Sequences

Proof Let W be any neighborhood of f(l). Since (xn) converges to l: V = f-1(W) is a neighborhood of l Thus, there exists N ϵ N s.t. n ≥ N implies xnϵ f-1(W) n ≥ N implies yn = f(xn) ϵ W. Hence: So (yn) converges to f(l).

VxNnN,N(l),V n∈⇒≥∈∃∈∀ V

WyNnN,N(f(l)),W n ∈⇒≥∈∃∈∀ VQED

INITIAL TOPOLOGY

Week 1 Lecture 1.6

Definition: Initial Topology

Let X and Yi be topological spaces (i ϵ I) Let fi : X→Yi be given mappings. We can equip X with a topology that makes every fi continuous. If everything else fails, the discrete topology will work! We call initial topology the coarsest one that works. We note it σ(X, {fi, i ϵ I}).

Example

Let X=R, Y=R and f be defined by f(x) = 0 if x≤0 f(x) = 1 if x>0

Equip Y with the usual topology. What is the initial topology on X for f? σ(X,{f}} = { } Ø

, ]-∞,0] , ]0, +∞[ , ]-∞,+∞[

Initial Topology and Sequences

Proposition Let X be a set. Let Yi be topological spaces (i ϵ I, finite or not). Let fi : X→Yi be given mappings. Let (xn) be a sequence of X. In the topology σ(X, {fi, i ϵ I}),

(xn) converges to x if and only if for all i ϵ I, fi(xn) converges to fi (x) in Yi

Initial Topology and Sequences

Proof Direct statement: Mappings fi are continuous for topology σ(X,{fi,iϵI}) We proved earlier that if (xn) converges to l then fi(xn) converges to fi(l).

Initial Topology and Sequences

Proof Converse: Let U be a neighborhood of x. We can suppose U is a finite intersection of inverse images of Vi

where Vi is a neighborhood of fi(x) in Yi. There exist Ni ϵ N s.t. n ≥ Ni implies fi(xn) ϵ Vi. Let N be the largest Ni (there is a finite number of Ni) Then n ≥ N implies xn ϵ U. QED

WHAT A TOPOLOGY “SEES” AND DOES NOT “SEE”

Week 1 Lecture 1.7

The “eyes” of a Topology

X = { 1 , 2 , 3 , 4, 5 } T = { Ø , {1,2} , {5} , {1,2,5} , {1,2,3,4} , X } 1 2 3 4 5

The topology does not distinguish between 1 and 2 nor between 3 and 4.