Let T: X X be the linear operator defined as K = 4. if N is even, K = , if N is odd, %3D Tx(t) = a2) with for any polynomial x(1) = Eoat living in X. That is, T returns the sum of the coefficients of the terms with even power of t. %3D Show that T is unbounded.
Let T: X X be the linear operator defined as K = 4. if N is even, K = , if N is odd, %3D Tx(t) = a2) with for any polynomial x(1) = Eoat living in X. That is, T returns the sum of the coefficients of the terms with even power of t. %3D Show that T is unbounded.
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
![(a) Let X be the space of all real valued polynomials of any degree in the closed interval
[0, 1]. That is, x e X if and only if there exist Ne N and ao, a..ay E R such
that
x(t) = ao +aịt +...+ aN-tN-1 + anT = a
a t',
for every r e [0, 1].
For any x e X written as above, consider X with the the norm given by
||| = max lal.
That is, the norm of x is equal to the maximum of the absolute value of the coeffi-
cients a, j = 1,...,N.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad4d02b6-78f1-4c0a-a713-3b04a8400050%2Fc03881ce-a130-49fc-b40e-86db5285a7f0%2F2x1evjpo_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Let X be the space of all real valued polynomials of any degree in the closed interval
[0, 1]. That is, x e X if and only if there exist Ne N and ao, a..ay E R such
that
x(t) = ao +aịt +...+ aN-tN-1 + anT = a
a t',
for every r e [0, 1].
For any x e X written as above, consider X with the the norm given by
||| = max lal.
That is, the norm of x is equal to the maximum of the absolute value of the coeffi-
cients a, j = 1,...,N.
Transcribed Image Text:ii. Let T : X X be the linear operator defined as
K = N.
K = , if N is odd,
if N is even,
Tx(1) =
a2)
with
for any polynomial x(1) = Eo at living in X. That is, T returns the sum of
the coefficients of the terms with even power of t.
!3!
Show that T is unbounded.
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