espect to some norm on X, but discontinuous with respect to another orm on X. To illustrate this, let X = Coo, the space of all scalar equences which have only finitely many nonzero members. Consider ne linear map fo(x)= x(1) + x(2)+..., x € X. he map fo is continuous with respect to the norm || ||1, since |fo(z)| ≤ Σ|x(j)| = ||*||₁, *€ X. j=1 ut fo is discontinuous with respect to the norm ||||2, since I₁ = 1,1/2,..., 1/n, 0, 0,...) € Coo and 1 -hile fo(In) = j=1 imilarly, consider the linear map f₁(x) = 8 WI [6) |f₁(x)| ≤ |x(j)| ≤ = as 818. 'hen f₁ is continuous with respect to the norm ||||2, since 1/2 1/2 - (2) ¹ (EGP) ¹² ||n||0o = 1, while fi(n) = * Ε.Χ. ut fi is discontinuous with respect to the norm || ||, since for n = (1, ..., 1, 0, 0, ...), where 1 appears n times, we have j=1 as 88.
espect to some norm on X, but discontinuous with respect to another orm on X. To illustrate this, let X = Coo, the space of all scalar equences which have only finitely many nonzero members. Consider ne linear map fo(x)= x(1) + x(2)+..., x € X. he map fo is continuous with respect to the norm || ||1, since |fo(z)| ≤ Σ|x(j)| = ||*||₁, *€ X. j=1 ut fo is discontinuous with respect to the norm ||||2, since I₁ = 1,1/2,..., 1/n, 0, 0,...) € Coo and 1 -hile fo(In) = j=1 imilarly, consider the linear map f₁(x) = 8 WI [6) |f₁(x)| ≤ |x(j)| ≤ = as 818. 'hen f₁ is continuous with respect to the norm ||||2, since 1/2 1/2 - (2) ¹ (EGP) ¹² ||n||0o = 1, while fi(n) = * Ε.Χ. ut fi is discontinuous with respect to the norm || ||, since for n = (1, ..., 1, 0, 0, ...), where 1 appears n times, we have j=1 as 88.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 20E: Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ]...
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Transcribed Image Text:(b) A linear map on a linear space X may be continuous with
respect to some norm on X, but discontinuous with respect to another
norm on X. To illustrate this, let X = coo, the space of all scalar
sequences which have only finitely many nonzero members. Consider
the linear map
fo(x) = x(1) + x(2)+..., x € X.
The map fo is continuous with respect to the norm || ||1, since
x(j)| = ||||₁, 2€ X.
j=1
But fo is discontinuous with respect to the norm ||||2, since n =
(1,1/2,..., 1/n, 0, 0, ...) € coo and
while
|fo(x)| ≤
||xn|| ²/2 Σ
Σ-
Similarly, consider the linear map
=
fo(Tn):
1
|f₁(x)| ≤|x(j)|
j=1
/
6
x(j)
fr(x) = Ĉª(2), x € X.
j=1
Then f₁ is continuous with respect to the norm ||||2, since
1/2
1/2
< (Σ;)" (Στ)
j=1
as 120.
But fi is discontinuous with respect to the norm || |, since for
(1,...,1,0,0,...), where 1 appears n times, we have
||n||∞ = 1, while fi(n)
while_fi(zn) = £} →
as 8-x.
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