Properties of $L^{\infty}$

Question

I'm trying to get a better grasp on the idea of $L^{\infty}$. What are the implications if we are given that $f \in L^{\infty}$? Also, how do we write $\|f\|_{\infty}$ in terms of the inf of a set of measure $0$? And is $L^\infty$ a normed (vector space) because we can define $\|f\|_{\infty}$ on it ( it follows the triangle inequality etc.), but what should we show if we wanted to show $L^\infty$ is a vector space?