Kenneth Kuttler

404 CHAPTER 14. ANALYSIS OF LINEAR TRANSFORMATIONS

9. If X is a finite dimensional normed vector space and A,B∈L (X ,X) such that ||B||<||A|| and A−1 exists, can it be concluded that

∣∣∣∣A−1B∣∣∣∣ < 1? Either give a counter

example or a proof.

10. Let X be a vector space with a norm ||·|| and let V = span(v1, · · · ,vm) be a finitedimensional subspace of X such that {v1, · · · ,vm} is a basis for V. Show V is a closedsubspace of X . This means that if wn→ w and each wn ∈V, then so is w. Next showthat if w /∈V,

dist(w,V )≡ inf{||w− v|| : v ∈V}> 0

is a continuous function of w and

|dist(w,V )−dist(w1,V )| ≤ ∥w1−w∥

Next show that if w /∈ V, there exists z such that ||z|| = 1 and dist(z,V ) > 1/2. Forthose who know some advanced calculus, show that if X is an infinite dimensionalvector space having norm ||·|| , then the closed unit ball in X cannot be compact.Thus closed and bounded is never compact in an infinite dimensional normed vectorspace.

11. Suppose ρ (A)< 1 for A ∈L (V,V ) where V is a p dimensional vector space havinga norm ||·||. You can use Rp or Cp if you like. Show there exists a new norm |||·|||such that with respect to this new norm, |||A|||< 1 where |||A||| denotes the operatornorm of A taken with respect to this new norm on V ,

|||A||| ≡ sup{|||Ax||| : |||x||| ≤ 1}

Hint: You know from Gelfand’s theorem that

||An||1/n < r < 1

provided n is large enough, this operator norm taken with respect to ||·||. Show thereexists 0 < λ < 1 such that

(Aλ

)< 1.

You can do this by arguing the eigenvalues of A/λ are the scalars µ/λ where µ ∈σ (A). Now let Z+ denote the nonnegative integers.

|||x||| ≡ supn∈Z+

∣∣∣∣∣∣∣∣An

λnx

∣∣∣∣∣∣∣∣First show this is actually a norm. Next explain why

|||Ax||| ≡ λ supn∈Z+

∣∣∣∣∣∣∣∣An+1

λn+1x

∣∣∣∣∣∣∣∣≤ λ |||x||| .

12. Establish a similar result to Problem 11 without using Gelfand’s theorem. Use anargument which depends directly on the Jordan form or a modification of it.

13. Using Problem 11 give an easier proof of Theorem 14.4.6 without having to useCorollary 14.4.5. It would suffice to use a different norm of this problem and thecontraction mapping principle of Lemma 14.4.4.

4049.10.11.12.13.CHAPTER 14. ANALYSIS OF LINEAR TRANSFORMATIONSIf X is a finite dimensional normed vector space and A, B € Y (X,X) such that ||B]| <\|A|| and A~! exists, can it be concluded that ||A~'B]| < 1? Either give a counterexample or a proof.Let X be a vector space with a norm |]-|| and let V = span(v1,--- ,vm) be a finitedimensional subspace of X such that {v1,--- , vm} is a basis for V. Show V is a closedsubspace of X. This means that if w, — w and each w, € V, then so is w. Next showthat if w ¢ V,dist (w,V) =inf{||w—v|]:veEV}>0is a continuous function of w and|dist (w,V) —dist(w1,V)| < ||wi —w]|Next show that if w ¢ V, there exists z such that ||z|| = 1 and dist(z,V) > 1/2. Forthose who know some advanced calculus, show that if X is an infinite dimensionalvector space having norm ||-||, then the closed unit ball in X cannot be compact.Thus closed and bounded is never compact in an infinite dimensional normed vectorspace.Suppose p (A) < 1 forA € &(V,V) where V is a p dimensional vector space havinga norm ||-||. You can use R? or C? if you like. Show there exists a new norm |||-|||such that with respect to this new norm, |||A||| < 1 where |||A]|| denotes the operatornorm of A taken with respect to this new norm on V,|||A||| = sup {|||Aa|||: ||[ae||] < 1}Hint: You know from Gelfand’s theorem thatAn <r<lprovided n is large enough, this operator norm taken with respect to ||-||. Show thereexists 0 < A < 1 such thatA>~)<l.e(z)You can do this by arguing the eigenvalues of A/A are the scalars /A where pb €o (A). Now let Z+ denote the nonnegative integers.A”x||| = sup ||=72zHealt sup || 37First show this is actually a norm. Next explain whyn+1|[|Aa||] =A sup <A||I2||].neZy.qn xEstablish a similar result to Problem 11 without using Gelfand’s theorem. Use anargument which depends directly on the Jordan form or a modification of it.Using Problem 11 give an easier proof of Theorem 14.4.6 without having to useCorollary 14.4.5. It would suffice to use a different norm of this problem and thecontraction mapping principle of Lemma 14.4.4.