(g) Consider E=C([0,1]) induced with the uniform norm and consider the line ar form: E→R defined by 4(f) 2(1) = [ ²³ ƒ (1)d² − [ ₁ f(t)dt- f(t)dt, fEE. i. Show that is continuous and calculate |||| ii. Show that if y(f)| = 1 with ||f||∞ = 1, then f(t): = What do you conclude? 1, 0
(g) Consider E=C([0,1]) induced with the uniform norm and consider the line ar form: E→R defined by 4(f) 2(1) = [ ²³ ƒ (1)d² − [ ₁ f(t)dt- f(t)dt, fEE. i. Show that is continuous and calculate |||| ii. Show that if y(f)| = 1 with ||f||∞ = 1, then f(t): = What do you conclude? 1, 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
part f g ii and iii functionnal analysis
![Problem 1. Let E be a real vector space, F be a closed subspace of E, and a € E\F.
Set d(a, F) inf{||ax||; x = F}.
1. Show that if F ha a finite dimension then there exists xo F such that d(x, F) =
||x-xo||.
2. Let be nonzero linear continuous form on E and a € E such that p(a) 0. We
consider F Ker p.
(a) Show that ||||d(a, Ker 4)
|p(a)|
(b) Show that (u+ta)|
||u + ta||
Ker +Ra).
(c) Deduce that |||| =
<
|y(a)|
d(a, kerp)'
|p(a)|
d(a, Ker )
Vu € Ker p. (Hint: Recall that E=
(d) Show that if uo Ker such that d(a, Ker p) = ||a - roll then there exists
To E such that|ro|| = 1 and |(ro)| = ||||-
(e) Let zo E so that oto+ta with v Ker . Suppose that ||zo|| = 1 and
€
(0)|||||. Show that told(a, Ker y) = 1.
(f) Deduce from the previous part that if zo € E such that ||zo|| = 1 and y(x) =
|||| then there exists uo € Ker such that d(a, Ker y) = ||a - xo||-
(g) Consider E= C([0, 1]) induced with the uniform norm and consider the line ar
form: ER defined by
9($) = [*¹ f(t)dt = ['f(t}dt,_ ƒ = E.
i. Show that is continuous and calculate ||||.
ii. Show that if y(f)|=1 with ||f|| = 1, then f(t):
What do you conclude?
[1, 0<t<
-1, <t<1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f7691fd-6982-45b0-a564-3f369d4d1266%2Fe64c92a4-6671-4f34-b303-56fcec5d6bcd%2F9ue10jm_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 1. Let E be a real vector space, F be a closed subspace of E, and a € E\F.
Set d(a, F) inf{||ax||; x = F}.
1. Show that if F ha a finite dimension then there exists xo F such that d(x, F) =
||x-xo||.
2. Let be nonzero linear continuous form on E and a € E such that p(a) 0. We
consider F Ker p.
(a) Show that ||||d(a, Ker 4)
|p(a)|
(b) Show that (u+ta)|
||u + ta||
Ker +Ra).
(c) Deduce that |||| =
<
|y(a)|
d(a, kerp)'
|p(a)|
d(a, Ker )
Vu € Ker p. (Hint: Recall that E=
(d) Show that if uo Ker such that d(a, Ker p) = ||a - roll then there exists
To E such that|ro|| = 1 and |(ro)| = ||||-
(e) Let zo E so that oto+ta with v Ker . Suppose that ||zo|| = 1 and
€
(0)|||||. Show that told(a, Ker y) = 1.
(f) Deduce from the previous part that if zo € E such that ||zo|| = 1 and y(x) =
|||| then there exists uo € Ker such that d(a, Ker y) = ||a - xo||-
(g) Consider E= C([0, 1]) induced with the uniform norm and consider the line ar
form: ER defined by
9($) = [*¹ f(t)dt = ['f(t}dt,_ ƒ = E.
i. Show that is continuous and calculate ||||.
ii. Show that if y(f)|=1 with ||f|| = 1, then f(t):
What do you conclude?
[1, 0<t<
-1, <t<1
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