Chapter 1
Functional analysis
Xiao-Min Huang
mahuangxm@gdut.edu.cn
Nov 2, 2020
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Chapter 1
Outline
1
Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
What we will learn this chapter
What is the purpose of functional analysis?
What is a topology (拓扑)?
What is an open set and a closed set?
What is a compact set (紧集)?
What is convergence? {un = n1 }
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
What we will learn this chapter
What is the purpose of functional analysis?
What is a topology (拓扑)?
What is an open set and a closed set?
What is a compact set (紧集)?
What is convergence? {un = n1 }
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
What we will learn this chapter
What is the purpose of functional analysis?
What is a topology (拓扑)?
What is an open set and a closed set?
What is a compact set (紧集)?
What is convergence? {un = n1 }
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
What we will learn this chapter
What is the purpose of functional analysis?
What is a topology (拓扑)?
What is an open set and a closed set?
What is a compact set (紧集)?
What is convergence? {un = n1 }
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
What we will learn this chapter
What is the purpose of functional analysis?
What is a topology (拓扑)?
What is an open set and a closed set?
What is a compact set (紧集)?
What is convergence? {un = n1 }
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
What we will learn this chapter
What is the purpose of functional analysis?
What is a topology (拓扑)?
What is an open set and a closed set?
What is a compact set (紧集)?
What is convergence? {un = n1 }
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
What we will learn this chapter
What is the purpose of functional analysis?
What is a topology (拓扑)?
What is an open set and a closed set?
What is a compact set (紧集)?
What is convergence? {un = n1 }
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
What we will learn this chapter
What is the purpose of functional analysis?
What is a topology (拓扑)?
What is an open set and a closed set?
What is a compact set (紧集)?
What is convergence? {un = n1 }
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
What we will learn this chapter
What is continuity?
What is the initial topology?
Trick your calculus instructor: every function can be
continuous?
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
What we will learn this chapter
What is continuity?
What is the initial topology?
Trick your calculus instructor: every function can be
continuous?
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
What we will learn this chapter
What is continuity?
What is the initial topology?
Trick your calculus instructor: every function can be
continuous?
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
What we will learn this chapter
What is continuity?
What is the initial topology?
Trick your calculus instructor: every function can be
continuous?
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Definition: Topology
Let X be a set and T be a family of subsets of X . T is called a
topology on X if:
The empty set ∅ and X are elements of T ;
Any union of elements of T is in T ;
Any finite intersection of elements of T is in T .
(X , T ) is a topology space (拓扑空间).
Elements of T are called open sets.
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Definition: Topology
Let X be a set and T be a family of subsets of X . T is called a
topology on X if:
The empty set ∅ and X are elements of T ;
Any union of elements of T is in T ;
Any finite intersection of elements of T is in T .
(X , T ) is a topology space (拓扑空间).
Elements of T are called open sets.
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Definition: Topology
Let X be a set and T be a family of subsets of X . T is called a
topology on X if:
The empty set ∅ and X are elements of T ;
Any union of elements of T is in T ;
Any finite intersection of elements of T is in T .
(X , T ) is a topology space (拓扑空间).
Elements of T are called open sets.
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Quiz
X = {1, 2, 3, 4, 5}
T = {{1, 2}, {3, 4}}
Is T a topology?
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Quiz
X = {1, 2, 3, 4, 5}
Which ones are topologies?
T = {∅, {1, 2}, {3, 4}, {1, 2, 3, 4}, X }
T = {∅, {1, 2}, {2, 3}, {1, 2, 3, 4}, X }
T = {∅, {1, 2}, {2, 3}, {2}, X }
T = {∅, {1, 2}, {2, 3}, {2}, {1, 2, 3}, X }
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
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 ∅ 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 .
Note: 2 and 3 cannot be permuted!
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
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 ∅ 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 .
Note: 2 and 3 cannot be permuted!
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
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 ∅ 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 .
Note: 2 and 3 cannot be permuted!
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Definition: Topology
A set X is always a topological space...
The trivial topology (平庸拓扑): Tt = {∅, X }.
The discrete topology (离散拓扑): Td = {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 .
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Quiz
X = {1, 2, 3, 4, 5}
T1 = {∅, {1, 2}, {1, 2, 3, 4}, X }
T2 = {∅, {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.
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
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.
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Definition: Closed sets
Complements of open sets are called closed sets.
Examples:
X = {1, 2, 3, 4, 5}
T = {∅, {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}
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Let X be a topological space.
Let x ∈ X .
The set U is called a neighborhood of x if
there exists an open set V s.t.
i. x ∈ V ;
ii. V ⊆ U.
The set of neighborhood of x is noted V (x ).
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
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 ∈ I)
whose union contains K ,
one can find a finite number of these open sets
such that their union contains K .
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Example: Compactness
Let X = R be equipped with the usual topology.
K = (0, 1] is not compact.
Ui = (1/i, 2) for (i ∈ N \ {0}).
The union contains K .
But a finite number of these open sets is
not enough to cover K .
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Example: Compactness
Let X = R be equipped with the usual topology.
Is K = [0, +∞) is a compact set?
Ui = (−1, i) for (i ∈ N \ {0}).
The union contains K .
But a finite number of these open sets is
not enough to cover K .
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
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 has a supremum M
A is not empty (it contains 0)
A is bounded by 1
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Example: Compactness
Let X = R be equipped with the usual topology.
K = [0, 1] is compact.
[0, M] can be cover 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)
It leads to a contradiction, therefore it is wrong.
Thus M=1.
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
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
∀V ∈ V (l), ∃N ∈ N, n ≥ N ⇒ xn ∈ V .
(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.
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Hausdorff Spaces
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.
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
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 stronger (larger/ finer),
It is "easier" for f to be continuous.
If X is equipped with the discrete topology,
any mapping is continous.
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Continuous Mappings and Sequences
The chosen definitions "work well":
Proposition
Let X and Y be two topological spaces.
Let f : X → Y 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 .
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Proof
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Proof
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Definition: Initial Topology
Let X and Yi be topological spaces (i ∈ 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 work.
We note it σ(X , {fi , i ∈ I}).
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
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 }) = {∅, (−∞, 0], (0, +∞), (−∞, +∞)}.
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Proposition
Let X be a set.
Let Yi be topological spaces (i ∈ I, finite or not).
Let fi : X → Yi be given mappings.
In the topology σ(X , {fi , i ∈ I}),
(xn ) converges to x if and only if
for all i ∈ I, fi (xn ) converges to fi (x ) in Yi .
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Proof
Proof
Direct statement:
Mappings fi are continuous for topology
σ(X , {fi , i ∈ I}).
We proved earlier that if (xn ) converges to x then
fi (xn ) converges to fi (x ).
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Proof
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 ∈ N s.t. n ≥ Ni implies fi (xn ) ∈ Vi .
Let N be the largest Ni
(there is a finite number of Ni ).
Then n ≥ N implies xn ∈ U.
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Chapter 1
What we will learn this chapter
Open sets and topology
Compact sets
Convergence and Continuity
Initial Topology
Thank you for your attention!
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