Preliminaries

1. Zorn’s Lemma

Relation S: an arbitary set R SXS R is called a relation on S

Partial Order A relation R is called a partial order on S if (1) if xRy and yRz, then xRz (2) xRx (3)’ if xRy and yRx, then x=y (not necessary) If R is a partial order, then R is usually denoted by ≦ or <, and (S, ≦ ) is called a poset.

Totally Ordered Let (S, ≦ ) be a poset and Q is called totally ordered, if for any either x ≦ y or y ≦ x hold.

Inductive poset Let (S, ≦ ) be a poset and an element c of S is called an upper bound of A if x ≦ c (S, ≦ ) is an inductive poset if every totally ordered subset has an uper bound.

Maximal Let (S, ≦ ) be a poset. An element m of S is called maximal if whenever and m ≦ y, then y ≦ m (if (3)’ hold, then y=m)

Zorn’s Lemma Let (S, ≦ ) be a inductive poset, then S has an maximal element. see Dunfud and Schwartz, Linear operations Chapter 1 or Kelly, General topology

2. Vector spaces

Linearly Independent Let E be a vector space over R or C A subset B of E is linearly independent if every finite subset of B is linearly independent

Proposition Let E be a vector space over R or C Let S be the family of all linearlly independent subset of E partially ordered by inclusion i.e. for a ≦ b means Then S is inductive.

Proof of Proposition Let Q be any totally ordered subset of S and let then, and c is an upper bound of Q

Hamel basis p.1 Let E be a vector space over R or C Let S be the family of all linearlly independent subset of E partially ordered by inclusion By previous proposition, S is inducitive By Zorn’s Lemma, S has a maximal element,say b. b is called a Hamel basis of E.

Hamel basis p.2 Let E be a vector space over R or C Let b be a Hamel basis of E

Hypersubspace p.1 Let E be a vector space over R or C A vector subspace (v.s.s) H of E is called a Hypersubspace of E if codim H =1, i.e. E/H is one dimensional Note:, [x]=x+H [x]+[y]=[x+y] λ[x]=[λx]

Hypersubspace p. 2 Let E be a vector space over R or C Let H be a Hypersubspace of E if and only if there is a linear functional l such that H=ker l Proof: Necessary:Let E/H is one dimensional, then

Hypersubspace p. 3 for any then λ(x) is linear functional on E and kerλ=H sufficiency:dim(Im l )=1,then codimH=1,then H is Hyperspace.

Convex is convex if and only if for any and

3. Sublinear functionals

Sublinear functional Let E be a real vector space a function p:E → R is called a subliear functional if (1) It is positive homogeneous. i.e. (2) It is subadditive i.e.

Exercise Let E be a real vector space p:E → R is subliear functional Show that p(0)=0 Proof: Suppose that p(0) ≠0, Since 2p(0)=p(2 ·0)=p(0), 2p(0)=p(0) then 2=1, which is impossible. Therefore p(0)=0

Example for sublinear Let S be an arbitary nonempty set and let E be the space of all real bounded functions on S. (E=B(S) ) p:E → R is defined by then p is sublinear functional

Interior point Let, a point is called an interior point of C

Exercise Let p be a sublinear functional on E,and. Show that (1)every point of c is an interior point (2)Both c and are convex.

Solution of Exercise p.1

Solution of Exercise p.2

Solution of Exercise p.3

Solution of Exercise p.4

Minkowski gauge Theorem Let K be a convex set in E with 0 being its interior point. Define a function

Proof of Minkowski gauge Theorem p.1

Proof of Minkowski gauge Theorem p.2

Minkowski gauge function of K is called the Minkowski gauge function of K

Exercise Show that

Solution of Exercise(1)

Solution of Exercise(2) p.1

Solution of Exercise(2) p.2

Solution of Exercise(2) p.3

4. Metric spaces

Metric space (X, ρ) is called a metric space if (1) ρ(x,y) ≧ 0,= hold if and only if x=y (2) ρ(x,y)= ρ(y,x) (3) ρ(x,z) ≦ ρ(x,y) +ρ(x,z)

Topology (X, ρ) is a metric space The family of all open subsets of X is called the topology of the metric space

Cauchy sequence

Complete p.1 A metric space is called complete if every Cauchy sequence converges in X. If X is not complete, one can construct a complete metric space in the following way. Let C b e the set of all Cauchy sequences in X.

Metric Completion p.1 Two Cauchy sequence in C are called equivalent if,denoted by x~y

Metric Completion p.2 Let ( 證明不難，但是挺麻煩的 )

Metric Completion p.3 Let T be the mapping from X to

Metric Completion p.4 Then

5. Normed vector spaces

Normed Vector Space p.1 Let E be a real or complex vector space.

Banach Space

Examples for Banach Space

Contruct a Normed Vector Space p.1

Contruct a Normed Vector Space p.2 (1)

Contruct a Normed Vector Space p.3 (2) (3)

Contruct a Normed Vector Space p.3

6. Lower semicontinuity

Lower Semicontiuous

Exercise 4.(1)

Proof of Exercise 4.(1) p.1

Proof of Exercise 4.(1) p.2

Exercise 4.(2)

Proof of Exercise 4.(2) p.1

Proof of Exercise 4.(2) p.2

Exercise 4.(3)

Proof of Exercise 4.(3) p.1

CHAPTER ONE Hahn-Banach Theorem Introduction to Theory of cojugate of convex functions

I.1 Analytic Form of Hahn- Banach Theorem Extension of linear functional

Theorem I.1 Hahn-Banach, analytic form Algebraic dual of G

Proof of Theorem I.1 p. 1

Proof of Theorem I.1 p. 2

Proof of Theorem I.1 p. 3

Proof of Theorem I.1 p. 4

Simple Application

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Recall ?

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Corollary I.2

Proof of Corollary I.2

Corollary I.3

Proof of Corollary I.3

Duality map