2. Preliminaries

3. 1. Zorn’s Lemma

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

5. 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.

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

7. 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.

8. 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)

9. 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

10. 2. Vector spaces

11. 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

12. 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.

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

14. 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.

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

16. 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]

17. 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

18. 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.

19. Convex is convex if and only if for any and

20. 3. Sublinear functionals

21. 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.

22. 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

23. 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

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

25. 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.

26. Solution of Exercise p.1

27. Solution of Exercise p.2

28. Solution of Exercise p.3

29. Solution of Exercise p.4

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

31. Proof of Minkowski gauge Theorem p.1

32. Proof of Minkowski gauge Theorem p.2

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

34. Exercise Show that

35. Solution of Exercise(1)

36. Solution of Exercise(2) p.1

37. Solution of Exercise(2) p.2

38. Solution of Exercise(2) p.3

40. 4. Metric spaces

41. 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)

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

43. Cauchy sequence

44. 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.

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

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

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

48. Metric Completion p.4 Then

49. 5. Normed vector spaces

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

51. Banach Space

52. Examples for Banach Space

53. Contruct a Normed Vector Space p.1

54. Contruct a Normed Vector Space p.2 (1)

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

56. Contruct a Normed Vector Space p.3

57. 6. Lower semicontinuity

58. Lower Semicontiuous

59. Exercise 4.(1)

60. Proof of Exercise 4.(1) p.1

61. Proof of Exercise 4.(1) p.2

62. Exercise 4.(2)

63. Proof of Exercise 4.(2) p.1

64. Proof of Exercise 4.(2) p.2

65. Exercise 4.(3)

66. Proof of Exercise 4.(3) p.1

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

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

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

70. Proof of Theorem I.1 p. 1

71. Proof of Theorem I.1 p. 2

72. Proof of Theorem I.1 p. 3

73. Proof of Theorem I.1 p. 4

74. Simple Application

76. ?

77. ?

78. Recall ?

80. ?

83. Corollary I.2

84. Proof of Corollary I.2

86. Corollary I.3

87. Proof of Corollary I.3

89. Duality map