Page 209 - DMTH401_REAL ANALYSIS

Unit 15: Uniform Convergence of Functions

15.7 Review Questions Notes

n
ì 1, x Î [0, 1/ ]
1. Let I = [1, 2] and ( )= í . f converges pointwise to
f x
n n
n
î 0, x Î [1, 2/ , 1]
n
ì 1, x Î [0, 1/ ]
f x
f(x) = ( )= í . f but does not converge uniformly to f.
n 0, x Î [1, 2/ , 1] n
n
î
2. Let =I [0, 1] and f to from Figure 15.5. Then f converges to the zero function pointwise
n
n
and uniformly.
1
.
x
3. Let =I (0, ¥ and f n ( )= ( + n ) 2 Then f converges to ( ) =f x 0 pointwise and uniformly.
)
x
n
4. { f n n ¥ = 1 be a sequence of real-valued functions on [ , ]a b . If all f are all continuous and
}
n
f  f uniformly on [ , ]a b then the limit function f is continuous.
n
5. Let I n = ( - 1/3, n + 1/3), n Î  . Then { }I n n Î  covers  , but does not cover  or {1/2}.
n
-
+
Let =S {1} È (3 1/4, 3 1/4). Then { }I n n Î  is a cover for S. Moreover, { ,I I 3 is a finite
}
1
subcover. Consider another case where { }I n n Î  is a cover of ¥. Here { }I n n Î  has no finite
subcover.
n
6. Let I n = ( 1 1/ , 1 1/ ), n Î  \{1}. We see that the collection {(–3/4, 3/4)} is a finite
-
-
n
+
-
subcover for set =S [ 17/24, 17/24]. Now, consider set = ( 1,1). Is { }I n n ³ 2 a cover for
-
S
S? The answer is positive and { }I n n ³ 2 has no finite subcover.
Answers: Self Assessment
1. uniform convergence 2. converges pointwise
3. S   if S  È aÎA I a 4. subcover of { }I a aÎA
15.8 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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