Page 234 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Equicontinuous at a Point: The family F is equicontinuous at a point x X if for every  > 0,
0
there exists a  > 0 such that d(f(x ), f(x)) <  for all ƒ  F and all x such that d(x , x) < . The family
0 0
is equicontinuous if it is equicontinuous at each point of X.
Uniform Boundedness: The uniform boundedness principle states that a pointwise bounded
family of continuous linear operators between Banach spaces is equicontinuous.

18.9 Review Questions

1. Explain the Equicontinuity and Families of Equicontinuous.
2. Describe the Properties of equicontinuous.
3. Discuss the Equicontinuity and uniform convergence.

4. Describe Equicontinuity families of linear operators.
5. Explain the Equicontinuity in topological spaces.
6. Define Stochastic equicontinuity.

Answers: Self Assessment

1. uniform boundedness principle 2. equicontinuous at a point
3. uniformly convergent subsequence 4. the Banach-Steinhaus theorem

5. Stochastic equicontinuity

18.10 Further Readings

Books Walter Rudin: Principles of Mathematical Analysis (3rd edition), Ch. 2, Ch. 3.
(3.1-3.12), Ch. 6 (6.1 - 6.22), Ch.7(7.1 - 7.27), Ch. 8 (8.1- 8.5, 8.17 - 8.22).
G.F. Simmons: Introduction to Topology and Modern Analysis, Ch. 2(9-13),
Appendix 1, p. 337-338.
Shanti Narayan: A Course of Mathematical Analysis, 4.81-4.86, 9.1-9.9, Ch.10,Ch.14,
Ch.15(15.2, 15.3, 15.4)
T.M. Apostol: Mathematical Analysis, (2nd Edition) 7.30 and 7.31.
S.C. Malik: Mathematical Analysis.
H.L. Royden: Real Analysis, Ch. 3, 4.

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