We denote by ∞ the space of all bounded sequences (an)∞n=1.For example,(1, −2, 1, −2, 1, −2,...) ∈ ∞ .Define addition and scalar multiplication by(an)∞n=1 + (bn)∞n=1 = (an + bn)∞n=1 andc(an)∞n=1 = (can)∞n=1 .a) Let ||(an)∞n=1|| = supn|an|. Show that || · || is a norm on ∞.b) Show that ∞ is complete with respect to this norm.In other words, prove ∞ is a Banach space. (22) Wedenote by t the space of all bounded sequences (4n)For example,(1, -2,1, -2, 1,-2, ...) E .Define addition and scalar multiplication by(a,)+ (b,), (a,+ b,) and%3Dc(an) = (ca,=1a) Let (a) = supla]. Show that | | is a norm on%3DTIb) Show that is complete with respect to this norm.In other words, prove (* is a Banach space.

Answered: Image /qna-images/answer/1c20de0e-a9d9-4b11-aad9-9c19aeff2c91.jpg

(22) Wedenote by the space of all bounded sequences (a,)-1 For example, (1, -2,1, –2, 1,–2, ...) E . Define addition and scalar multiplication by (a,)+ (b,), = (an+ b„) and %3D c(an) = (ca,=1: a) Let |(a,.)l = supla,]. Show that | | is a norm on %3D b) Show that ( is complete with respect to this norm. In other words, prove ( is a Banach space.

(22) Wedenote by the space of all bounded sequences (a,)-1 For example, (1, -2,1, –2, 1,–2, ...) E . Define addition and scalar multiplication by (a,)+ (b,), = (an+ b„) and %3D c(an) = (ca,=1: a) Let |(a,.)l = supla,]. Show that | | is a norm on %3D b) Show that ( is complete with respect to this norm. In other words, prove ( is a Banach space.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.2: Vector Spaces

Problem 48E: Let R be the set of all infinite sequences of real numbers, with the operations...

See similar textbooks