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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

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
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Question
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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
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