Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
. Suppose f is C-analytic in D(0; 1) and maps the unit circle into itself. Show then that f maps the entire disc onto itself. [Hint: Use the Maximum-Modulus Theorem to show that f maps D(0; 1) into itself. Then apply the previous exercise to conclude that the mapping is onto.]

. Suppose f is C-analytic in D(0; 1) and maps the unit circle into itself. Show then that f maps the entire disc onto itself. [Hint: Use the Maximum-Modulus Theorem to show that f maps D(0; 1) into itself. Then apply the previous exercise to conclude that the mapping is onto.]

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.4: Spanning Sets And Linear Independence
Problem 76E: Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are...
See similar textbooks
icon
Related questions
Question
100%

TRANSCRIBE THE FOLLOWING TEXT IN DIGITAL FORMAT AND EXPLAIN EACH STEP

4. Suppose f is C-analytic in D(0; 1) and maps the unit circle into itself. Show then that f maps the entire disc onto itself. [Hint: Use the Maximum-Modulus Theorem to show that f maps D(0; 1) into itself. Then apply the previous exercise to conclude that the mapping is onto.]
Transcribed Image Text:4. Suppose f is C-analytic in D(0; 1) and maps the unit circle into itself. Show then that f maps the entire disc onto itself. [Hint: Use the Maximum-Modulus Theorem to show that f maps D(0; 1) into itself. Then apply the previous exercise to conclude that the mapping is onto.]
het D denote D(0,1) and cl (D) the closed disc {12/51} The maximum modulus theorem gives that f maps D into itself. So the issue is showing that it is onto Proof: By the open mapping theorem PCD) is open, So f (D)n D is open. Since cl CD) is Compact, f (c (D)) is also compact. Hence f(cl(D)) D is a relatively closed subset of D. gimce f takes {121=1} to {121=1}, f (c (D)) DD= f(D) AD. Hence f(D) AD is open, closed and nonempty Subset of D. Hence by connection connectedness of D you have that f(D) AD = D which means f is onto as well.
Transcribed Image Text:het D denote D(0,1) and cl (D) the closed disc {12/51} The maximum modulus theorem gives that f maps D into itself. So the issue is showing that it is onto Proof: By the open mapping theorem PCD) is open, So f (D)n D is open. Since cl CD) is Compact, f (c (D)) is also compact. Hence f(cl(D)) D is a relatively closed subset of D. gimce f takes {121=1} to {121=1}, f (c (D)) DD= f(D) AD. Hence f(D) AD is open, closed and nonempty Subset of D. Hence by connection connectedness of D you have that f(D) AD = D which means f is onto as well.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra
College Algebra
Algebra
ISBN:
9781938168383
Author:
Jay Abramson
Publisher:
OpenStax
SEE MORE TEXTBOOKS