**Question **# Problem 5.1. Let S be a set and let F(S) be the vector space of allfunctions f : S → F. Show that F(S) is a vector space over F with theoperations defined in class. 2. Let [a, b] be a closed interval in R. Show that C([a, b]), the collectionof continuous functions f : [a, b] → R, is a subspace of the real vectorspace F([a, b]). \text { 3. Show that } \mathcal{P}\left(\mathbb{F}^{\mathrm{r}}\right) \text { and } \mathcal{P}_{n}\left(\mathrm{~F}^{\mathrm{r}}\right) \text { is a subspace of } F\left(\mathrm{~F}^{\mathrm{r}}\right) \text {. } 4. Suppose W1, . .,Wm are vector spaces over F. Verify that W1×·×W,mis a vector space with the operations defined in class.