Inner Product $p\neq 2$ I am liking know if exist a any generalization for inner product in the set $L^p(\mathbb{R}^2)$ , for any $p\geq 1$.

On any real Banach space $X$ one can define semi-inner products 
$$\begin{split}
\langle x,y\rangle_+ &= \sup \{f(x):f\in X^*, \|f\|=1, f(y)=\|y\|\} \\
\langle x,y\rangle_- &= \inf\{f(x):f\in X^*, \|f\|=1, f(y)=\|y\|\}
\end{split}
\tag1$$
There is an equivalent definition in terms of directional derivative of the norm. 
The semi-inner products  are not linear in $y$ unless we are in a Hilbert space. In general, they are not linear in $x$ either: $\langle x,y\rangle_+$ is convex and $\langle x,y\rangle_-$ is concave. But on smooth Banach spaces, such as $L^p$ for $1<p<\infty$, the set of functionals $f$ in (1) consists of just one element. This implies $\langle x,y\rangle_+ = \langle x,y\rangle_-$ and subsequently, the semi-inner product $\langle x,y\rangle$ becomes linear in $x$. 
The Cauchy-Schwarz inequality holds in this form: 
$$-\|x\|\,\|y\|\le \langle x,y\rangle_- \le \langle x,y\rangle_+ \le \|x\|\,\|y\| \tag2$$
Reference: Nonlinear Functional Analysis by Deimling.
