Counterexamples on Uniform Convergence
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"The features of the book include An overview of important concepts and theorems on uniform convergence, Well-organized coverage of the majority of the topics on uniform convergence studied in analysis courses, An original approach to the analysis of important results on uniform convergence based on counterexamples, Additional exercises at varying levels of complexity for each topic covered in the book & A supplementary Instructor's Solutions Manual containing complete solutions to all exercises, which is available via a companion website" Mathematical Reviews, Sept 2017More details
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Chapter 1. Conditions of Uniform Convergence
Example 1. A function f(x, y) defined on converges pointwise on X as y approaches y0, but this convergence is nonuniform on X
A sequence of functions converges (pointwise) on a set, but this convergence is nonuniform.
A series of functions converges (pointwise) on a set, but this convergence is nonuniform.
Example 2. A series of functions converges on X and a general term of the series converges to zero uniformly on X, but the series converges nonuniformly on X.
Example 3. A sequence of functions converges on X and there exists its subsequence that converges uniformly on X, but the original sequence does not converge uniformly on X.
Example 4. A function f(x, y) defined on converges on (a, b) as y approaches y0 and this convergence is uniform on any interval , but the convergence is nonuniform on (a, b).
A sequence of functions defined on (a, b) converges uniformly on any interval , but the convergence is nonuniform on (a, b)
A series of functions converges uniformly on any interval , but the convergence is nonuniform on (a, b).
Example 5. A sequence converges on X, but this convergence is nonuniform on a closed interval .
A series converges on X, but this series does not converge uniformly on a closed subinterval .
A function f(x, y) defined on has a limit for , but f(x, y) converges nonuniformly on a subinterval .
Example 6. A sequence converges on a set X, but it does not converge uniformly on any subinterval of X.
A series converges on X, but it does not converge uniformly on any subinterval of X.
Example 7. A series converges uniformly on an interval, but it does not converge absolutely on the same interval.
Example 8. A series converges absolutely on an interval, but it does not converge uniformly on the same interval.
Example 9. A series converges absolutely and uniformly on [a, b], but the series does not converge uniformly on [a, b].
Example 10. A series converges absolutely and uniformly on X, but there is no bound of the general term un(x) on X in the form , such that the series converges.
Example 11. A sequence fn(x) converges uniformly on X to a function f(x), but does not converge uniformly on X to f2(x).
Sequences fn(x) and gn(x) converge uniformly on X, but fn(x)gn(x) does not converge uniformly on X.
Example 12. Sequences fn(x) and gn(x) converge nonuniformly on X to f(x) and g(x), respectively, but converges to uniformly on X.
Example 13. A sequence converges uniformly on X, but fn(x) diverges on X.
A sequence converges uniformly on X, but fn(x) diverges on X.
A sequence converges uniformly on X and fn(x) converges on X, but the convergence of fn(x) is nonuniform.
Example 14. A sequence converges uniformly on X to 0, but neither fn(x) nor gn(x) converges to 0 on X.
Example 15. A sequence fn(x) converges uniformly on X to a function f(x), , , , but does not converge uniformly on X to .
Example 16. A sequence fn(x) is bounded uniformly on R and converges uniformly on , , to a function f(x), but the numerical sequence does not converge to .
Example 17. Suppose each function fn(x) maps X on Y and function g(y) is continuous on Y; the sequence fn(x) converges uniformly on X, but the sequence does not converge uniformly on X.
Suppose functions fn(x) map X on Y and function g(y) is continuous on Y; the sequence fn(x) converges nonuniformly on X, but the sequence converges uniformly on X
Example 18. A series converges uniformly on X, but the series does not converge uniformly on X.
Example 19. A series converges uniformly on X, but the series does not converge (even pointwise) on X.
Example 20. A series converges uniformly on X, but at least one of the series or does not converge uniformly on X.
Example 21. A series converges uniformly on X, but neither nor converges (even pointwise) on X.
Example 22. Series and converge nonuniformly on X, but converges uniformly on X.
Example 23. A series converges uniformly on X, but does not converge uniformly on X.
Both series and converge uniformly on X, but the series does not converge uniformly on X.
Example 24. A series converges uniformly on X, but does not converge (even pointwise) on X.
Both series and converge uniformly on X, but the series does not converge (even pointwise) on X.
Example 25. Both series and are nonnegative for , and one of these series converges uniformly on X, but another series does not converge uniformly on X.
Example 26. The partial sums of are bounded for , and the sequence vn(x) is monotone in n for each fixed and converges uniformly on X to 0, but the series does not converge uniformly on X.
Example 27. The partial sums of are uniformly bounded on X and the sequence vn(x) converges uniformly on X to 0, but the series does not converge uniformly on X.
Example 28. The partial sums of are uniformly bounded on X, and the sequence vn(x) is monotone in n for each fixed and converges on X to 0, but the series does not converge uniformly on X.
Example 29. The partial sums of are not uniformly bounded on X, and the sequence vn(x) is not monotone in n and does not converge uniformly on X to 0, but still the series converges uniformly on X.
A series diverges at each point of X, and a sequence vn(x) is not monotone in n and diverges on X, but still the series converges uniformly on X.
Example 30. A series converges on X, and a sequence vn(x) is monotone in n for each fixed and uniformly bounded on X, but the series does not converge uniformly on X.
Example 31. A series converges uniformly on X, and a sequence vn(x) is uniformly bounded on X, but the series does not converge uniformly on X.
Example 32. A series converges uniformly on X, and a sequence vn(x) is monotone in n for each fixed , but the series does not converge uniformly on X.
A series converges uniformly on X, and a sequence vn(x) is monotone and bounded in n for each fixed , but the series does not converge uniformly on X.
Example 33. A series does not converge uniformly on X, and a sequence vn(x) is not monotone in n and is not uniformly bounded on X, but still the series converges uniformly on X.
A series diverges at each point of X, and a sequence vn(x) is not monotone in n and is not bounded at each point of X, but still the series converges uniformly on X.
Chapter 2. Properties of the Limit Function: Boundedness, Limits, Continuity
Example 1. A sequence of bounded on X functions converges on X to a function, which is unbounded on X.
Example 2. A sequence of functions is not uniformly bounded on X, but it converges on X to a function, which is bounded on X.
Example 3. A sequence of unbounded and discontinuous on X functions converges on X to a function, which is bounded and continuous on X.
Example 4. A sequence of unbounded and discontinuous on X functions converges on X to an unbounded and discontinuous function, but the convergence is nonuniform on X.
A sequence of unbounded and continuous on X functions converges on X to an unbounded and continuous function, but the convergence is nonuniform on X.
Example 5. A sequence of unbounded and discontinuous on X functions converges on X to an unbounded and discontinuous function, but this convergence is uniform on X.
Example 6. A sequence of uniformly bounded on...
