Consider $R^2$ valued functions $f,g \in L^2([0,1],\mu, R^2)$ where $f=(f_1,f_2)$,$g=(g_1,g_2)$ and $\sqrt{\langle f, g\rangle}=\int_{[0,1]} (f_1(x)g_1(x)+f_2(x)g_2(x)) d\mu(x)$
Suppose for a given $g$ and $\alpha\in R$, there exists $f$ satisfying $\langle f, g\rangle=\alpha$,
Can I always find one such $f$ with $\langle f, g\rangle=\alpha$ further to be $f_1(x)=f_2(x)=f^*(x)$, for all $x$? or
Does the equation $\int_{[0,1]} f^*(x)[g_1(x)+g_2(x)] d\mu(x)=\alpha$ always allow a solution?
If now there exists $f$ such that $\langle f,g\rangle=\alpha>sup_{g'\in G} \langle f,g' \rangle$, where $G$ is a convex and compact subset of $L^2([0,1],R^2)$, could we have such $f$ with $f_1(x)\leq f_2(x)$ for all x?
Thanks.
We have to look at the hyperplanes $$ H_\alpha = \{ f \in L^2([0,1]; \def\R{\mathbf R}\R^2) \mid \def\<#1>{\left<#1\right>}\<f,g> = \alpha\} $$ and the subspace $$ U_\Delta = \{ f \in L^2([0,1]; \R^2) \mid f_1 - f_2 = 0 \}. $$ As $H_\alpha$ is a hyperplane, $U_\Delta$ is either parallel to it, or has non-empty intersection with it. The orthogonal subspace to $H_\alpha = f + H_0$ (where $f \in H_\alpha$) is $$ H_0^\bot = \R \cdot g $$ Now $U_\Delta$ and $H_\alpha$ are parallel if $H_0^\bot \bot U_\Delta$, that is $g \bot U_\Delta$, or $\int_{[0,1]} f(g_1+g_2)\,d\mu = 0$ for all $f \in L^2([0,1];\R)$, that is $g_1 + g_2 = 0$ or $g_1 - g_2 = 0$. In this case, $U_\Delta$ and $H_\alpha$ are parallel, and hence have no intersection or $U_\Delta \subseteq H_\alpha$, as $0 \in U_\Delta$, the latter happens iff $\alpha = 0$.
Summarizing, we can say: Such an $f$ exists iff $g_1 \ne -g_2$ or $\alpha = 0$.