Linear extension of a functional on closed unit ball.

Let $X$ be a normed linear space with closed unit ball $B$. Suppose the function $f: B\rightarrow [-1,1]$ has the property that whenever $x,y,x+y$ and $\lambda x$ belong to $B$, $f(x+y)=f(x)+f(y)$ and $f(\lambda x)=\lambda f(x)$. Show that $f$ is the restriction to $B$ of a linear functional on all of $X$.

I was wondering how to prove this proposition? Hahn-Banach theorem does not seem to work.Any help is appreciated.