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Let V denote the set of all differentiable real-valued functions defined on the real line. Prove that V is a vector space with the operations of addition and scalar multiplication as follows:

(f + g)(s) = f(s) + g(s) and (cf)(s) = c[f(s)]

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I know I have to prove this by checking the 8 axioms of vector spaces, but I'm very confused as to how I actually go about it in a formal mathematical proof. If someone could just prove one of them (say commutativity of addition a+b=b+a), I'm sure I'll be able to figure out the rest.

I'm just confused about proving something so fundamental. I keep thinking there are no operations I can perform without first proving what I'm trying to prove. GAH

(f + g)(s) = f(s) + g(s) and (cf)(s) = c[f(s)]

---------------

I know I have to prove this by checking the 8 axioms of vector spaces, but I'm very confused as to how I actually go about it in a formal mathematical proof. If someone could just prove one of them (say commutativity of addition a+b=b+a), I'm sure I'll be able to figure out the rest.

I'm just confused about proving something so fundamental. I keep thinking there are no operations I can perform without first proving what I'm trying to prove. GAH

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