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Chapter 4 Hilbert Space

Chapter 4 Hilbert Space. 4.1 Inner product space. Inner product. E : complex vector space. is called an inner product on E if. Inner product space. E : complex vector space. Show in next some pages. is an inner product on E. With such inner product E is called.

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Chapter 4 Hilbert Space

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  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 Show in next some pages 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.

  5. Schwarz Inequality E is an inner product space

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

  7. Example 1 for Inner product space Let For

  8. Example 2 for Inner product space Let For

  9. Example 3 for Inner product space Let For

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

  11. Exercise 1.1 (ii) Show that is complete

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

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

  14. 4.2 Geometry for Hilbert space

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

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

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

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

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

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

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

  22. Exercise 2.2 (iii)

  23. Exercise 2.2 (iv)

  24. Exercise 2.2 (v) conversely if then

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

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

  27. Exercise 2.2 (viii)

  28. Exercise 2.2 (ix)

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

  30. 4.3 Linear transformation

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