Kenneth Kuttler

13.7. EXERCISES 351

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

)< 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+1 x

∣∣∣∣∣∣∣∣ ≤ 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 13.6.6 without having to use Corol-lary 13.6.5. It would suffice to use a different norm of this problem and the contractionmapping principle of Lemma 13.6.4.

14. A matrix A is diagonally dominant if |aii| >∑

j ̸=i |aij | . Show that the Gauss Seidelmethod converges if A is diagonally dominant.

15. Suppose f (λ) =∑∞

n=0 anλn converges if |λ| < R. Show that if ρ (A) < R where A is

an n× n matrix, then

f (A) ≡∞∑

n=0

anAn

converges in L (Fn,Fn) . Hint: Use Gelfand’s theorem and the root test.

13.7. EXERCISES 35110.11.12.13.14.15.Let X be a vector space with a norm ||-|| and let V = span (v1,--- , Um) be a finitedimensional subspace of X such that {v1,--+ ,Um} 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||:v EV} >0is a continuous function of w and\dist (w,V) — dist (wi, V)| < ||jwi — 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 for A € £(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,||| Al] = sup {||| Ax|]] + [I[xI]] < 1}Hint: You know from Gelfand’s theorem thatAr <r<lprovided n is large enough, this operator norm taken with respect to ||-||. Show thereexists 0 < \ < 1 such that A~)]<i.(5)You can do this by arguing the eigenvalues of A/ are the scalars js/ where fz € a (A).Now let Z; denote the nonnegative integers.I|[x||| = supneZynmye *First show this is actually a norm. Next explain why||| Ax||] = A sup < || |x|].neZyyrti 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 13.6.6 without having to use Corol-lary 13.6.5. It would suffice to use a different norm of this problem and the contractionmapping principle of Lemma 13.6.4.A matrix A is diagonally dominant if |aj:| > }) jz; |aij|. Show that the Gauss Seidelmethod converges if A is diagonally dominant.Suppose f (A) = °°, and” converges if |A| < R. Show that if p(A) < R where A isann X n matrix, thenf(A) = Soa,”n=0converges in £(F”,F”). Hint: Use Gelfand’s theorem and the root test.