a) Let X = {fe C[0,1]:f (0) = 0} Y = {g ex :[g (t) dt = 0} %3D Prove that Y is a proper subspace of X. Is Y a closed subspace of X? Justify your answer.
a) Let X = {fe C[0,1]:f (0) = 0} Y = {g ex :[g (t) dt = 0} %3D Prove that Y is a proper subspace of X. Is Y a closed subspace of X? Justify your answer.
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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![1. a)
Let X =
{f e C[0,1]:f (0) = 0}
1
Y =
{ g e x:[g(t)dt = 0}
Prove that Y is a proper subspace of X. Is Y a closed subspace of X? Justify your
answer.
b)
Let x
L'[0,1] and x = x (t) = t. Find x for p = 4 and o.
X
c)
Let E be a subset of a normed space X, Y = span E and a e X. Show that a e Y if and
only if f (a):
= 0 whenever fe X'and f = 0 everywhere on E.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5687772d-b32b-40f6-87ba-be13f259e06b%2Ffd8f3e3f-34d2-4bc3-a002-d2bda1af3aae%2Ft14va5q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. a)
Let X =
{f e C[0,1]:f (0) = 0}
1
Y =
{ g e x:[g(t)dt = 0}
Prove that Y is a proper subspace of X. Is Y a closed subspace of X? Justify your
answer.
b)
Let x
L'[0,1] and x = x (t) = t. Find x for p = 4 and o.
X
c)
Let E be a subset of a normed space X, Y = span E and a e X. Show that a e Y if and
only if f (a):
= 0 whenever fe X'and f = 0 everywhere on E.
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