-E= T
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
![E =
le (0) =
with the inner product
{u: [0, 7] → RN | u is absolutely continuous,
u(T), ù € L²([0, 7]. RN))
[[(i(1), v (1)) + (u(1), v(1))]dr
for all u, v € E where (...) denotes the inner product in RN. The corresponding norm
is defined by
AU. VYTE=
||u||E= [(lù(1³² +\u(1)1³²)dr. V u € E.
For every u, ve E, we define
=S² eQ(¹) [(ù (1), v(1)) + (A(1)u(1), v(t))]dt,
4. V Y=
and we observe that, by the assumptions (A1) and (A2), it defines an inner product in
E. Then, E is a separable and reflexive Banach space with the norm
||u|| =< u, u > ² Vue E.
||ul|oo ≤S|| u ||
where ||ul|∞ = maxic[0,71 |u(t)].
The question is why is norm
defined in this way, please clarify?
(2)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7123a2b5-b9e1-43ef-a62b-ab6cb4b75ea2%2Fb0d660b4-dafd-4521-b37a-c28dafc2ca8c%2F3rjnjvp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:E =
le (0) =
with the inner product
{u: [0, 7] → RN | u is absolutely continuous,
u(T), ù € L²([0, 7]. RN))
[[(i(1), v (1)) + (u(1), v(1))]dr
for all u, v € E where (...) denotes the inner product in RN. The corresponding norm
is defined by
AU. VYTE=
||u||E= [(lù(1³² +\u(1)1³²)dr. V u € E.
For every u, ve E, we define
=S² eQ(¹) [(ù (1), v(1)) + (A(1)u(1), v(t))]dt,
4. V Y=
and we observe that, by the assumptions (A1) and (A2), it defines an inner product in
E. Then, E is a separable and reflexive Banach space with the norm
||u|| =< u, u > ² Vue E.
||ul|oo ≤S|| u ||
where ||ul|∞ = maxic[0,71 |u(t)].
The question is why is norm
defined in this way, please clarify?
(2)
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