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Question 1.11.6: Consider the vector space C[−1,1] of all continuous function...

Consider the vector space C[−1,1] of all continuous functions on the interval [−1,1]. Let H be the set of all functions with the property that f (−1) = f (1) = 2. Determine whether or not H is a subspace of C[−1,1].

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The set H does not satisfy any of the three required properties of subspaces, so any one of these suffices to show that H is not a subspace. In particular, the zero function z(t )=0 does not have the property that z(−1)=2,and thus the zero function from C[−1,1] does not lie inH, soH is not a subspace.
We could also observe that any scalar multiple of a function whose value at t =−1 and t = 1 is 2 will result in a new function whose value at these points is not 2; similarly, the sum of two functions whose values at t =−1 and t =1 are 2 will lead to a new function whose values at these points is 4. These facts together show that H is not closed under scalar multiplication, nor under addition.

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