I am attempting to show the triangle inequality holds for the following norm, which is defined as: $||f|| = \bigg( \int_0^1 (|f|^2 + |f'|^2) \ dx \bigg)^{1/2} \\$.
My attempt thus far; I am confused on where to go next? Any help would be great.
$\begin{align*} ||f+g||^2 &= \int_0^1 |f+g|^2+ |f'+g'|^2 \ dx \\ &= \int_0^1 |f|^2 + 2|fg| +|g|^2 |f'|^2 + 2|f'g'| + |g'|^2 \ dx \\ &= \int_0^1 |f|^2 + |f'|^2 \ dx + \int_0^1 |g|^2 + |g'|^2 \ dx + 2 \int_0^1 |fg|+|f'g'| \ dx \\ \end{align*}$