Problem 2. Let a, b ɛ R, with a < b. Consider the set B[a, b)] of all bounded functions from the interval [a, b] into R. Show that B[a,b] is a vector space over R when endowed with pointwise addition and scalar multiplication.
Problem 2. Let a, b ɛ R, with a < b. Consider the set B[a, b)] of all bounded functions from the interval [a, b] into R. Show that B[a,b] is a vector space over R when endowed with pointwise addition and scalar multiplication.
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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solve problem 2 with explanation asa and get multiple upvotes.
![• Problem 1. Let (V, (·|·)) be an inner product space. Show, for all v, w E V,
that |(v|w)| < (v|v)!/2(w\w)'/2. (Hint: Let t e R be arbitrary and consider
(v + tw|v+tw)).
• Problem 2. Let a, b e R, with a < b. Consider the set B[a,b] of all bounded
functions from the interval [a, b] into R. Show that B[a,b] is a vector space
over R when endowed with pointwise addition and scalar multiplication.
• Problem 3. Let a, b e R, with a < b. With B[a,b] as above, define
||f | := sup{|f(s)| : s € [a,b]}
Show that || · |∞ is a norm on B[a,b].
(ƒ € B[a, b).
• Problem 4: Let (M, d) be a metric space and let ACM be non-empty.
Prove that the following are equivalent:
(1) A is bounded.
(2) There exists a D >0 so that, for every a,b ɛ A, we have d(a, b) < D.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39537d6d-3d3e-4c05-b0a6-92a7be108579%2Ffea1c09b-d2f0-42df-9db0-eef0c0fbc314%2Fqqw72uo_processed.png&w=3840&q=75)
Transcribed Image Text:• Problem 1. Let (V, (·|·)) be an inner product space. Show, for all v, w E V,
that |(v|w)| < (v|v)!/2(w\w)'/2. (Hint: Let t e R be arbitrary and consider
(v + tw|v+tw)).
• Problem 2. Let a, b e R, with a < b. Consider the set B[a,b] of all bounded
functions from the interval [a, b] into R. Show that B[a,b] is a vector space
over R when endowed with pointwise addition and scalar multiplication.
• Problem 3. Let a, b e R, with a < b. With B[a,b] as above, define
||f | := sup{|f(s)| : s € [a,b]}
Show that || · |∞ is a norm on B[a,b].
(ƒ € B[a, b).
• Problem 4: Let (M, d) be a metric space and let ACM be non-empty.
Prove that the following are equivalent:
(1) A is bounded.
(2) There exists a D >0 so that, for every a,b ɛ A, we have d(a, b) < D.
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