1. Chapter 4 Hilbert Space

2. 4.1 Inner product space

3. Inner product E : complex vector space is called an inner product on E if

4. Inner product space E : complex vector space is an inner product on E With such inner product E is called inner product space. If we write,then is a norm on E and hence E is a normed vector space. Show in next some pages

5. Schwarz Inequality E is an inner product space

7. Triangular Inequality for ∥ ．∥ E is an inner product space

9. Example 1 for Inner product space Let For

10. Example 2 for Inner product space Let For

11. Example 3 for Inner product space Let For

12. Exercise 1.1 (i) For Show that and hence is absolutely convergent

14. Exercise 1.1 (ii) Show that is complete

17. Hilbert space An inner product space E is called is complete Hilbert space if is a Hilbert space of which

18. Exercise 1.2 Define real inner product space and real Hilbert space.

19. 4.2 Geometry for Hilbert space

20. Theorem 2.1 p.1 E: inner product space M: complete convex subset of E Let then the following are equivalent

21. Theorem 2.1 p.2 (1) satisfing (1) and (2). (2) Furthermore there is a unique

26. Projection from E onto M The map of Thm 1 is called the projection from E onto M. y is the unique element in M which satisfies (1) defined by tx=y, where and is denoted by

27. Corollary 2.1 Let M be a closed convex subset of a Hilbert has the following properties: space E, then

29. Convex Cone A convex set M in a vector space is called a convex cone if

30. Exercise 2.2 (i) Let M be a closed convex cone in a Hilbert Put Show that space E and let I being the identity map of E.

32. Exercise 2.2 (ii) ( t is positive homogeneous)

34. Exercise 2.2 (iii)

36. Exercise 2.2 (iv)

38. Exercise 2.2 (v) then conversely if

40. Exercise 2.2 (vi) M is a closed vector subspace of E. Show that In the remaining exercise, suppose that

42. Exercise 2.2 (vii) both t and s are continuous and linear

44. Exercise 2.2 (viii)

46. Exercise 2.2 (ix)

48. Exercise 2.2 (x) such that x=y+z tx and sx are the unique elements

50. 4.3 Linear transformation

51. We consider a linear transformation from vector space Y over the same field R or C. a normed vector space X into a normed

52. Exercise 1.1 T is continuous on X if and only if T is continuous at one point.

54. Theorem 3.1 T is continuous if and only if there is a such that

56. Theorem 3.3 Riesz Representation Theorem Let X be a Hilbert space and Furthermore the map such that then there is is conjugate linear and

58. 4.4 Lebesgue Nikondym Theorem

59. Indefinite integral of f Let Suppose that and f a Σ –measurable function on Ω be a measurable space has a meaning; then the set function defined by is called the indefinite integral of f.

60. Property of Indefinite integral of f ν is σ- additive i.e. if is a disjoint sequence, then

61. Absolute Continuous ν is said to be absolute continuous w.r.t μ if

62. Theorem 4.1 Lebesgue Nikodym Theorem with Suppose that νis absolute continuous w.r.t. μ, then there is a unque such that Furthermore

67. 4.5 Lax-Milgram Theorem

68. Sesquilinear p.1 Let X be a complex Hilbert space. is called sesquilinear if

69. Sesquilinear p.2 B is called bounded if there is r>0 such that B is called positive definite if there is ρ>0 s.t.

70. Theorem 5.1 The Lax-Milgram Theorem p.1 Let X be a complex Hilbert space and B a a bounded, positive definite sesquilinear functional on X x X, then there is a unique bounded linear operator S:X → X such that and

71. Theorem 5.1 The Lax-Milgram Theorem p.2 Furthermore exists and is bounded with

76. 4.7 Bessel Inequality and parseval Relation

77. Propositions p.1 Let be an orthogonal system in a Hilbert space X, and let U be the closed vector subspace generated by Letbe the orthogonal projection onto U and where

78. Proposition (1)

80. Proposition (2)

82. Proposition (3) For each k and x,y in X

84. Proposition (4) For any x,y in X

86. Proposition (5) Bessel inequality

88. Proposition (6) An orthonormal system is called complete if U=X ( Parseval relation)

90. Separable A Hilbert space is called separable if it contains a countable dense subset

91. Theorem 7.1 A saparable Hilbert space is isometrically isomorphic either to for some n or to