Is : $||(a+b)^2 || \le ||a||^2+2||ab||+||b||^2$ true in any form when a,b, are elements of a space? Let $E$ be a space and $a,b$ be elements of it, 
$$||(a+b)^2 || \le ||a||^2+2||ab||+||b||^2 $$, where $|| \cdot ||$ is most of the time a sum norm ,  then also a  form for integrals shows up some times : 
$$\int |a(t)+b(t)|^2dt \le \int |a(t)|^2dt +2 \int |a(t)b(t)|dt +\int |b(t)|^2dt $$
How to prove that this is true?

$(a+b)^2 = a^2 + 2ab + b^2$ in any ring.
$|x+y+z| \leq |x| + |y| + |z|$ for any norm.
Just put the two together.
