Exercise 14

Where $S$ is a set, the functions $f\colon S\to\mathbb{R}$ form a vector space under the natural operations: the sum $f+g$ is the function given by $f+g\,(s)=f(s)+g(s)$ and the scalar product is $r\cdot f\,(s)=r\cdot f(s)$. What is the dimension of the space resulting for each domain?

• (a)  $S=\text{$\{\hskip 1.00006pt1\hskip 1.00006pt\}$}$     (b)  $S=\text{$\{\hskip 1.00006pt1,2\hskip 1.00006pt\}$}$     (c)  $S=\text{$\{\hskip 1.00006pt1,\ldots,n\hskip 1.00006pt\}$}$