Let 1 < p <∞o and a = = (k) CK be a sequence. We define Dr = (kæk)k¹ w¹P = {r € | Dr € PP} and ||||₁p = |x||p+ ||Dx||p² Then || ||1p is a norm on wlp (you do not have to prove this). Prove that . a) (w¹P, ||- ||1p) is a separable Banach space. Hint: Identify w¹P with a subset of IP X IP b) Every bounded set M in (w¹P, ||- ||1p) is relatively compact in IP. Hint: Which criteria of relatively compactness for sets in IP do you already know?
Let 1 < p <∞o and a = = (k) CK be a sequence. We define Dr = (kæk)k¹ w¹P = {r € | Dr € PP} and ||||₁p = |x||p+ ||Dx||p² Then || ||1p is a norm on wlp (you do not have to prove this). Prove that . a) (w¹P, ||- ||1p) is a separable Banach space. Hint: Identify w¹P with a subset of IP X IP b) Every bounded set M in (w¹P, ||- ||1p) is relatively compact in IP. Hint: Which criteria of relatively compactness for sets in IP do you already know?
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:A further sequence space
Let 1 < p <∞o and a := = (%) CK be a sequence. We define Dz = (kæk)
w¹P = {x € | Dx EP} and ||||₁p = |||| + ||D||p
Then || 1.p is a norm on wlp (you do not have to prove this). Prove that
a) (w¹P, || ||1p) is a separable Banach space.
.
Hint: Identify w¹P with a subset of IP X IP
b) Every bounded set M in (w¹,||||1p) is relatively compact in IP.
Hint: Which criteria of relatively compactness for sets in IP do you already know?
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