Clearify I understood the question (best aproximation) Define the $p$ norm for $1<p<\infty$ $$
\|x\|_{p}=\left(\sum_{i=1}^{n}\left|x_{i}\right|^{p}\right)^{1 / p}
$$
Find all the best approximations of $(1, 1)$ by vectors in span {$(1, −1)$}with $||\cdot||_p$ on $\mathbb {C^2}$
First of all why they say approximations? I thought approximation is a unique value.
second do I need to solve this equation :$$<(1,1) - \alpha (1,-1),(1,1)>=0$$
$$<(1,1),(1,1)> - \alpha <(1,1),(1,-1)>=0$$
and from here just open it according to the definition of $||\cdot||_p$?

Let
$$e= (1,1)- \alpha(1,-1)$$
be the difference or error between the vector $(1,1)$ with a vector $\alpha(1,-1)\in \mbox{Span} \{(1,-1)\}$. For best approximations the length of the error should be minimimum. That is you have to find $\alpha$ such that
$$\|e\|_p = \|(1,1)-\alpha(1,-1)\|_p=\bigg((1-\alpha)^p +(1+\alpha)^p\bigg)^{\frac{1}{p}},$$ is minimum.
