Page 259 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes 23.2 Summary

 Let X be a linear space over the field of complex numbers C. An inner product on X is a
mapping from X × X C which satisfies the following conditions:

(i) ( x + y, z) = (x, z) + (y, z) x, y, z X and , C.

(ii) (x, y) = (y, x)
(iii) (x, x) 0, (x, x) = 0 x = 0

 A complete inner product space is called a Hilbert space.

23.3 Keywords

Hilbert Space: A complete inner product space is called a Hilbert space.
Inner Product Spaces: Let X be a linear space over the field of complex numbers C. An inner
product on X is a mapping from X × X C which satisfies the following conditions:
(i) ( x + y, z) = (x, z) + (y, z), x, y, z X and , C. (Linearity property)

(ii) (x, y) = (y, x) (Conjugate symmetry)
(iii) (x, x) 0, (x, x) = 0 x = 0

23.4 Review Questions

n
1. For the special Hilbert space  , use Cauchy’s inequality to prove Schwarz’s inequality.
2
n
2. Show that the parallelogram law is not true in  (n > 1).
2
3. If x, y are any two vectors in a Hilbert space H, then prove that
2
2
2
2
4 (x, y) = x + y – x – y + i x + iy – i x – iy .
4. If B is complex Banach space whose norm obeys the parallelogram law, and if an inner
product is defined on B by
2
2
2
2
4 (x, y) = x + y – x – y + i x + iy – i x – iy ,
then prove that B is a Hilbert space.
23.5 Further Readings

Books Bourbaki, Nicolas (1987), Topological vector Spaces, Elements of Mathematics, BERLIN:
Springer – Verlag.
Halmos, Paul (1982), A Hilbert space Problem Book, Springer – Verlag.

Online links www.math-sinica.edu.tw/www/file_upload/maliufc/liu_ch04.pdf

mathworld.wolfram.com>Calculus and Analysis > Functional Analysis

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