2. Suppose that V and W are vector spaces over F. Consider the cartesian product V x W, with vector addition and scalar multiplication defined by (V1, W1) + (V2, w2) = (V1 + V2, W1 + w2) and c(V1, W1) = (cv1, cW1) for every (v1, W1), (v2, w2) E V x W and ce F. a) Show that V x W is a vector space over F. b) Suppose that || - ||v is a norm on V and || - ||w is a norm on W. Show that |(v, w)|| = ||v||v + ||w||w defines a norm on V x W. c) Show that a sequence (vn, Wn)nEN in V x W converges to (v, w) E V x W as n o if and only if the sequence (Vn)n€N in V converges to v E V and the sequence (w,),neN in W converges to w e W asn - 00. d) Comment on Cauchy sequences.
2. Suppose that V and W are vector spaces over F. Consider the cartesian product V x W, with vector addition and scalar multiplication defined by (V1, W1) + (V2, w2) = (V1 + V2, W1 + w2) and c(V1, W1) = (cv1, cW1) for every (v1, W1), (v2, w2) E V x W and ce F. a) Show that V x W is a vector space over F. b) Suppose that || - ||v is a norm on V and || - ||w is a norm on W. Show that |(v, w)|| = ||v||v + ||w||w defines a norm on V x W. c) Show that a sequence (vn, Wn)nEN in V x W converges to (v, w) E V x W as n o if and only if the sequence (Vn)n€N in V converges to v E V and the sequence (w,),neN in W converges to w e W asn - 00. d) Comment on Cauchy sequences.
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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Transcribed Image Text:2. Suppose that V and W are vector spaces over F. Consider the cartesian product V x W, with vector
addition and scalar multiplication defined by
(V1, W1) + (v2, w2) = (V1 + v2, W1 + w2)
and
c(V1, W1) = (cv1, cw1)
for every (v1, w1), (v2, w2) E V x W and c € F.
a) Show that V x W is a vector space over F.
b) Suppose that || - ||v is a norm on V and || ||w is a norm on W. Show that
|(v, w)|| = ||v||v + ||w||w
defines a norm on V x W.
c) Show that a sequence (vn, Wn)nEN in V xW converges to (v, w) E V x W as n 00 if and only
if the sequence (vn)nEN in V converges to vEV and the sequence (wn)nEN in W converges to
w e W as n 00.
d) Comment on Cauchy sequences.
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