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So my question is: Determine whether the set equipped with the given operations is a vector space. For those that are not vector spaces identify the vector space axioms that fail. The set of all real-valued functions f defined everywhere on the real line and such that f(9) = 0, with the operations (f+g)(x) = f(x) + g(x) (kf)(x) = kf(x)

So based on the above I know that the set is closed under addition and under scalar multiplication. Is that enough to say it is a vector space or no?

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    Do you know that the set of real valued functions with point wise operations is a vector space? If you do, after you have shown that your set is not empty, it is a subspace and thus is a vector space itself. – 2017-01-12
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    No I do not, how would I show that? – 2017-01-12
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    Then you could also verify the vector space axioms for your set. It is basically the same work. – 2017-01-12
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    So I just read a quick wikipedia post on pointwise operations and the first part shows that addition works, but the multiplication is different. The wikipedia article says (f*g)(x) = f(x) * g(x), how would i transform my equation above to fit this. Also am I right with saying that this is a vector space? – 2017-01-12
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    that isn't a scalar multiplication. It is a multiplication which leads to a ring. You should use your definition as you want a vector space. – 2017-01-12

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