Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
2. Prove that every convergent sequence is Cauchy. 3. It is well-known that all norms on a finite dimensional space

2. Prove that every convergent sequence is Cauchy. 3. It is well-known that all norms on a finite dimensional space

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.4: Ordered Integral Domains
Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....
See similar textbooks
icon
Related questions
Question
2
Problems for Banach spaces 1. Check that the following functions on R² are norms: 3 a) || (T1, 12) ||p= (1" + |r2|" ), 1<p<o! 1/p b) || (x1, x2) ||00= max{ 1, 2. Prove that every convergent sequence is Cauchy. 3. It is well-known that all norms on a finite dimensional space are equivalent, that is if || : || and || · I|2 are two norms on a fine dimensional space, then there are positive constants c and C such that for every x c || x ||1<|| x |l2< C || x ||,: baloid or-on Find constants c and C for || (x1, x2) ||2 and || (x1, x2) ||. from problem 1. 4. Let || - ||1 and || - ||2 be two equivalent norms on a vector X. Prove that a sequence {Tn} in X converges to ro E X in || - ||1 if and only if it converges to o in || - ||2. 2,9one boon space t0 1ndt.2 ae Jibo a arher lo 5. Given two normed spaces X and Y equpped with norms || - ||x and ||- |ly respectively. The Cartesian product X x Yconsists of all pairs (x, y), T E X, y E Y. lisd on ei orsd nd2 lo Juiog xoai ns a). Prove that X x Y is a linear space. b). Prove that the following functions on X × Y are norms and that these norms are equivalent: || (x, y) ||1=|| x |x + || y ||y; || (x, y) |0= max{ || x ||x, || v ||r }. lon vam osd noite e sd eda twoda olamm a oviD a 6. Recall that G is a two dimensional space equipped with the norm I| (x1, 2) ||2= V[T|P + |x2l?. Let A be the following operator on 6: A(r1, r2) = (x1 + x2, 12). a). Prove that A is a linear operator. b). Prove that A is boinded and find it's norm
Transcribed Image Text:Problems for Banach spaces 1. Check that the following functions on R² are norms: 3 a) || (T1, 12) ||p= (1" + |r2|" ), 1<p<o! 1/p b) || (x1, x2) ||00= max{ 1, 2. Prove that every convergent sequence is Cauchy. 3. It is well-known that all norms on a finite dimensional space are equivalent, that is if || : || and || · I|2 are two norms on a fine dimensional space, then there are positive constants c and C such that for every x c || x ||1<|| x |l2< C || x ||,: baloid or-on Find constants c and C for || (x1, x2) ||2 and || (x1, x2) ||. from problem 1. 4. Let || - ||1 and || - ||2 be two equivalent norms on a vector X. Prove that a sequence {Tn} in X converges to ro E X in || - ||1 if and only if it converges to o in || - ||2. 2,9one boon space t0 1ndt.2 ae Jibo a arher lo 5. Given two normed spaces X and Y equpped with norms || - ||x and ||- |ly respectively. The Cartesian product X x Yconsists of all pairs (x, y), T E X, y E Y. lisd on ei orsd nd2 lo Juiog xoai ns a). Prove that X x Y is a linear space. b). Prove that the following functions on X × Y are norms and that these norms are equivalent: || (x, y) ||1=|| x |x + || y ||y; || (x, y) |0= max{ || x ||x, || v ||r }. lon vam osd noite e sd eda twoda olamm a oviD a 6. Recall that G is a two dimensional space equipped with the norm I| (x1, 2) ||2= V[T|P + |x2l?. Let A be the following operator on 6: A(r1, r2) = (x1 + x2, 12). a). Prove that A is a linear operator. b). Prove that A is boinded and find it's norm
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Elements Of Modern Algebra
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
SEE MORE TEXTBOOKS