Page 106 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 8: Bounded Linear Functional on the L -spaces
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8.3 Keywords Notes

Bounded Linear Functional on L -spaces: If x  and f is bounded linear functional on  , then
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p
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f has the unique representation of the form as an infinite series
f (x) = x f(e )
k k
k 1
Bounded Linear Functional: A linear functional f on a normed space N is said to be bounded if
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there is a constant k > 0 such that
|f (x)| k x , x N
1
Continuous Linear Functional: A linear functional f is continuous if given > 0 there exists >
0 so that
|f (x) – f (y)| whenever x – y .
Linear Functional: Let N be a normed space over a field R (or C). A mapping f : N R (or C)
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is called a linear functional on N if f ( x + y) = f (x) + f (y), x, y N and , R (or C).
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Norm: The norm of f  is given by
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q
f = |f(e )| q
k
k 1
8.4 Review Questions

1. Account for bounded linear functionals on L -space.
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2. State and prove different continuous linear functional theorems.
3. Describe approximation by continuous function.
4. How will you explain norms of bounded linear functional on L -space?
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5. What is Isometric Isomorphism?

8.5 Further Readings

Books Rudin, Walter (1991), Functional Analysis, Mc-Graw-Hill Science/Engineering/
Math
Kreyszig, Erwin, Introductory Functional Analysis with Applications, WILEY 1989.
T.H. Hilderbrandt, Transactions of the American Mathematical Society. Vol. 36,
No. = 4, 1934.

Online links www.math.psu.edu/yzheng/m597k/m597kLIII4.pdf
www.public.iastate.edu/…/Royden_Real_Analysis_Solutions.pdf

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