Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
1 S(x) = É +n²x"

1 S(x) = É +n²x"

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
100%
Q4 Real analysis by Walton rudin chp#7
is self-adjoint, we see that u e AR. If x, # x2, there exists feA such that f(x1) = 1, f(x2) = 0; hence 0 = u(x2) # u(x1) = 1, which shows that AR separates points on K. If x e K, then g(x) +0 for some g eA, and there is a complex number 2 such that ig(x) > 0; if f = ig, f = u + iv, it follows that u(x) > 0; hence AR vanishes at no point of K. Thus AR satisfies the hypotheses of Theorem 7.32. It follows that every real continuous function on K lies in the uniform closure of AR, hence lies in 3. If f is a complex continuous function on K, f = u +iv, then u e B, v E ®, hence f E B. This completes the proof. EXERCISES 1. Prove that every uniformly convergent sequence of bounded functions is uni- formly bounded. 2. If {f) and {g.) converge uniformly on a set E, prove that {f.+ gn} converges uniformly on E, If, in addition, {f) and {g.) are sequences of bounded functions, prove that (fag,) converges uniformly on E. 3. Construct sequences {f.), {gn} which converge uniformly on some set E, but such that {fagn) does not converge uniformly on E (of course, {fng.} must converge on E). 4. Consider f(x) = For what values of x does the series converge absolutely ? On what intervals does it converge uniformly? On what intervals does it fail to converge uniformly? Is f continuous wherever the series converges? Is f bounded? 166 PRINCIPLES OF MATHEMATICAL ANALYSIS 5. Let sin² (n+1 Show that {f} converges to a continuous function, but not uniformly. Use the series E f. to show that absolute convergence, even for all x, does not imply uni- form convergence. 6. Prove that the series converges uniformly in every bounded interval, but does not converge absolutely for any value of x. 7. For n= 1, 2, 3, ..., x real, put Show that {f.} converges uniformly to a function f, and that the equation f'(x)= lim f.(x) is correct if x #0, but false if x=0. 8. If (o (xS0), (x>0), I(x) = if (xn) is a sequence of distinct points of (a, b), and if E c.| converges, prove that the series
Transcribed Image Text:is self-adjoint, we see that u e AR. If x, # x2, there exists feA such that f(x1) = 1, f(x2) = 0; hence 0 = u(x2) # u(x1) = 1, which shows that AR separates points on K. If x e K, then g(x) +0 for some g eA, and there is a complex number 2 such that ig(x) > 0; if f = ig, f = u + iv, it follows that u(x) > 0; hence AR vanishes at no point of K. Thus AR satisfies the hypotheses of Theorem 7.32. It follows that every real continuous function on K lies in the uniform closure of AR, hence lies in 3. If f is a complex continuous function on K, f = u +iv, then u e B, v E ®, hence f E B. This completes the proof. EXERCISES 1. Prove that every uniformly convergent sequence of bounded functions is uni- formly bounded. 2. If {f) and {g.) converge uniformly on a set E, prove that {f.+ gn} converges uniformly on E, If, in addition, {f) and {g.) are sequences of bounded functions, prove that (fag,) converges uniformly on E. 3. Construct sequences {f.), {gn} which converge uniformly on some set E, but such that {fagn) does not converge uniformly on E (of course, {fng.} must converge on E). 4. Consider f(x) = For what values of x does the series converge absolutely ? On what intervals does it converge uniformly? On what intervals does it fail to converge uniformly? Is f continuous wherever the series converges? Is f bounded? 166 PRINCIPLES OF MATHEMATICAL ANALYSIS 5. Let sin² (n+1 Show that {f} converges to a continuous function, but not uniformly. Use the series E f. to show that absolute convergence, even for all x, does not imply uni- form convergence. 6. Prove that the series converges uniformly in every bounded interval, but does not converge absolutely for any value of x. 7. For n= 1, 2, 3, ..., x real, put Show that {f.} converges uniformly to a function f, and that the equation f'(x)= lim f.(x) is correct if x #0, but false if x=0. 8. If (o (xS0), (x>0), I(x) = if (xn) is a sequence of distinct points of (a, b), and if E c.| converges, prove that the series
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS