Let V be a vector space over a field F, and let W be a subset of V. We say that W is closed under addition if v1+v2 belongs to W whenever v1 and v2 both belong to W, and that it is closed under scalar multiplication if λv belongs to W whenever v belongs to W, for any scalar λ∈F. Show that W is a vector space (relative to the addition and multiplication induced on V) provided it is closed under addition and scalar multiplication. Use this,  to give a quick proof that the set of continuous, differentiable real-valued functions on R is a vector space over the real numbers under the usual operations

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Q: Please see image

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