Let $X\subseteq\mathbb{R}^{n}$ and let $L^{p}(X)$ denotes the Banach space of $p$-integrable functions $f: X\rightarrow\mathbb{R}$, with $1<p<\infty$. It is clear that the space $(L^p(X),\|\cdot\|_{p})$ is a strictly convex Banach space for $p > 1$, taking into account that $$ \|f\|_p = \left(\int_X |f(x) |^p dx \right)^{1/p}, $$ with the Lebesgue measure.
If we consider the space $L^p(X)\oplus_{p}L^p(X)$ with the norm $\|\cdot\|_{\oplus_{p}} : L^p(X)\oplus_p L^p(X)\rightarrow[0,\infty)$ defined by $$ \|(f,g)\|_{\oplus_{p}} = (\|f\|_p^p + \|g\|_p^p)^{1/p} $$ My question is prove that $(L^p(X)\oplus_{p} L^p(X),\|\cdot\|_{\oplus_p})$ is also strictly convex Banach space.
I hope this result is correct, but the problem is the terms raised to $p$. I am using the characterization that $h,k\in L^p(X)\oplus_{p} L^p(X)$, $h\neq k$ and such that $\|h\|_{\oplus_p} = 1 = \|k\|_{\oplus_p}$, then $\|h+k\|_{\oplus_p} < 1$.
Any idea?