Consider the set U = {f € Co (R)| limx→∞ f(x) = 0}. That is, U is the set of all continuous functions that go to zero as x goes to infinity. Is U a vector space? Check ALL 10 AXIOMS and clearly show which axioms U satisfies and which it does not. Then conclude whether or not U is a vector space. Include every step, if you ever ask yourself "should I also include this??" the answer is YES. Your solution must be clear, complete, and detailed. Hint: U is a set of functions. Don't confuse those functions with their limits.
Consider the set U = {f € Co (R)| limx→∞ f(x) = 0}. That is, U is the set of all continuous functions that go to zero as x goes to infinity. Is U a vector space? Check ALL 10 AXIOMS and clearly show which axioms U satisfies and which it does not. Then conclude whether or not U is a vector space. Include every step, if you ever ask yourself "should I also include this??" the answer is YES. Your solution must be clear, complete, and detailed. Hint: U is a set of functions. Don't confuse those functions with their limits.
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:Consider the set U = {ƒ € Co (R) | limx→∞ f(x)=0}.
That is, U is the set of all continuous functions that go to zero as x goes to infinity.
Is U a vector space? Check ALL 10 AXIOMS and clearly show which axioms U satisfies and which it does not. Then
conclude whether or not U is a vector space. Include every step, if you ever ask yourself "should I also include this??"
the answer is YES. Your solution must be clear, complete, and detailed. Hint: U is a set of functions. Don't confuse
those functions with their limits.
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