Definition of extreme point: A point $f$ in $L$ with $\|f\| = 1$ is called an extreme point of the unit ball if $$f = (1 - \alpha)f_1 + \alpha f_2, \, 0 < \alpha < 1, \, \|f_1\|, \|f_2\| \leq 1 \implies f_1 = f_2$$ Let $1 < p < \infty$. Show that every element $f$ of the unit ball of $L_p$ with $\|f\|=1$ is an extreme point.
I have seen proofs of the other direction, but there isn't any for this direction. Any help is appreciated! Thanks