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## Homework Statement

Let V = R

^{R}be the vector space of the pointwise functions from R to R. Determine whether or not the following subsets W contained in V are subspaces of V.

## Homework Equations

W = {f ∈ V : f(1) = 1}

W = {f ∈ V: f(1) = 0}

W = {f ∈ V : ∃f ''(0)}

W = {f ∈ V: ∃f ''(x) ∀x ∈ R}

## The Attempt at a Solution

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Note: I am doing Linear Algebra, but I have not yet done Calculus.

I know that to prove a subset W is a subspace, I must prove that the 0 vector belongs to W, that vector addition is closed, and scalar multiplication as well.

I am not sure how to go about proving these statements, however. My intuition tells me that the first subset is not a subspace and the second one is, and the other two I am not sure about. Nonetheless when I sit down to start proving, I'm very insecure about what I find.

For example, given (f+g)(1) = f(1) + g(1) = 2 ≠ 1, does this result mean vector addition is not closed?

In the case of the second problem, (f+g)(1) = f(1) + g(1) = 0 + 0 = 0, so addition is closed.

Similarly, for the second proof, (λf)(1) = λ1 ≠ 1 in my first problem, and in the second (λf)(1) = λ0 = 0.

Am I on the right track?

If someone could really help me with the third and fourth problems, I have a rudimentary understanding of derivatives and how they work and what the first and second derivatives can indicate about a function, but I do not know how to relate the statements in my problem to the concepts I know about subspaces.