vector space of real valued function on a given set 
well, $a$ is true I guess but not sure , $b$ is true as $\max\{f,g\}=\frac{1}{2}(|f-g|-|f+g|\}$
$c$ is also true as  $f\in V\Rightarrow f^2\in V$ so any polynomial expression will also be in $V$, could any one tell me my logic are correct or not? and help me to solve fully correctly?

a is indeed true, but you should give an argument for it (hint: $fg$ occurs in what kind of expression involving $f$, $g$ and $x \mapsto x^2$? With a bit of cleverness, you can make the excess $f^2$ and $g^2$ bits go away)
