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 vị and v2 both belong to W, and that it is closed under scalar multiplication if Av belongs to W whenever v belongs to W, for any scalar 1 E 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, along with your work in Question 1, 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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