Suppose f is a real continuous function on R', f.(t)=f(nt) for n=1, 2, 3, ..., and {S} is equicontinuous on [0, 1]. What conclusion can you draw about f?
Suppose f is a real continuous function on R', f.(t)=f(nt) for n=1, 2, 3, ..., and {S} is equicontinuous on [0, 1]. What conclusion can you draw about f?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Q15
![168 PRINCIPLES OF MATHEMATICAL ANALYSIS
14. Let f be a continuous real function on R' with the following properties:
Os f(1)<1, f(t+ 2) = f(t) for every t, and
(0
f(t)=
11
(SIS1).
Put 9(t) = (x(t), y(t)), where
x(1) =E2-"f(32n - 1f),
-E2-(3").
y(t) =
Prove that 0 is continuous and that O maps I= [0, 1] onto the unit square 12 c R?.
If fact, show that O maps the Cantor set onto 1?.
Hint: Each (xo, Yo) e 1? has the form
Xo =
Yo =
2-"azn
where each a, is 0 or 1. If
to =
' 3 -1-'(2a,)
show that f(3*to) = ar, and hence that x(to) = xo, y(to) = yo.
(This simple example of a so-called "space-filling curve" is due to I. J.
Schoenberg, Bull. A.M.S., vol. 44, 1938, pp. 519.)
15. Suppose f is a real continuous function on R', f.(1)=f(nt) for n=1, 2, 3, ..., and
{f) is equicontinuous on [0, 1]. What conclusion can you draw about f?
16. Suppose {f.} is an equicontinuous sequence of functions on a compact set K, and
{f} converges pointwise on K. Prove that {fa} converges uniformly on K.
17. Define the notions of uniform convergence and equicontinuity for mappings into
any metric space. Show that Theorems 7.9 and 7.12 are valid for mappings into
any metric space, that Theorems 7.8 and 7.11 are valid for mappings into any
complete metric space, and that Theorems 7.10, 7.16, 7.17, 7.24, and 7.25 hold for
vector-valued functions, that is, for mappings into any R*.
18. Let {f.} be a uniformly bounded sequence of functions which are Riemann-inte-
grable on [a, b], and put
F.(x) =
fa(t)
(a<xsb).
Prove that there exists a subsequence {F} which converges uniformly on [a, b].
19. Let K be a compact metric space, let S be a subset of C(K). Prove that S is compact
(with respect to the metric defined in Section 7.14) if and only if S is uniformly
closed, pointwise bounded, and equicontinuous. (If S is not equicontinuous,
then S contains a sequence which has no equicontinuous subsequence, hence has
no subsequence that converges uniformly on K.)
SEQUENCES AND SERIES OF FUNCTIONS 169
20. If f is continuous on [0, 1] and if
(n= 0, 1, 2, ...),
prove that f(x) = 0 on [0, 1]. Hint: The integral of the product of f with any](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa29dd1fd-ff33-4092-aedd-03e5d354d4ee%2F599e71d0-1eef-4a40-aa75-f4fc64be9cea%2Fje70ifl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:168 PRINCIPLES OF MATHEMATICAL ANALYSIS
14. Let f be a continuous real function on R' with the following properties:
Os f(1)<1, f(t+ 2) = f(t) for every t, and
(0
f(t)=
11
(SIS1).
Put 9(t) = (x(t), y(t)), where
x(1) =E2-"f(32n - 1f),
-E2-(3").
y(t) =
Prove that 0 is continuous and that O maps I= [0, 1] onto the unit square 12 c R?.
If fact, show that O maps the Cantor set onto 1?.
Hint: Each (xo, Yo) e 1? has the form
Xo =
Yo =
2-"azn
where each a, is 0 or 1. If
to =
' 3 -1-'(2a,)
show that f(3*to) = ar, and hence that x(to) = xo, y(to) = yo.
(This simple example of a so-called "space-filling curve" is due to I. J.
Schoenberg, Bull. A.M.S., vol. 44, 1938, pp. 519.)
15. Suppose f is a real continuous function on R', f.(1)=f(nt) for n=1, 2, 3, ..., and
{f) is equicontinuous on [0, 1]. What conclusion can you draw about f?
16. Suppose {f.} is an equicontinuous sequence of functions on a compact set K, and
{f} converges pointwise on K. Prove that {fa} converges uniformly on K.
17. Define the notions of uniform convergence and equicontinuity for mappings into
any metric space. Show that Theorems 7.9 and 7.12 are valid for mappings into
any metric space, that Theorems 7.8 and 7.11 are valid for mappings into any
complete metric space, and that Theorems 7.10, 7.16, 7.17, 7.24, and 7.25 hold for
vector-valued functions, that is, for mappings into any R*.
18. Let {f.} be a uniformly bounded sequence of functions which are Riemann-inte-
grable on [a, b], and put
F.(x) =
fa(t)
(a<xsb).
Prove that there exists a subsequence {F} which converges uniformly on [a, b].
19. Let K be a compact metric space, let S be a subset of C(K). Prove that S is compact
(with respect to the metric defined in Section 7.14) if and only if S is uniformly
closed, pointwise bounded, and equicontinuous. (If S is not equicontinuous,
then S contains a sequence which has no equicontinuous subsequence, hence has
no subsequence that converges uniformly on K.)
SEQUENCES AND SERIES OF FUNCTIONS 169
20. If f is continuous on [0, 1] and if
(n= 0, 1, 2, ...),
prove that f(x) = 0 on [0, 1]. Hint: The integral of the product of f with any
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