Let $X$ be a non-empty Borel subset of $\mathbb{R}$ and consider the finite-measure space $(X,\mathcal{B}(X),\mu)$. Fix $y,f^1,\dots,f^n \in L^2_{\mu}(X)$ and define the objective function $$ \begin{aligned} E:\mathbb{R}^n &\rightarrow (-\infty,\infty] \\ \beta &\mapsto \int_{x \in X} (y(x)-\sum_{i=1}^n \beta^if^i(x))^2 d\mu(x). \end{aligned} $$ Is $E$ a strictly lsc convex optimization problem over $\mathbb{R}^d$? If $X$ is discrete or it is equal to $\mathbb{R}$ then I know this, but what about in general?
Intuition: If we ensure that $L^2_{\mu}(X)$ containts the constant functions, then is this a convex optimization problem over the constraint set $$ \left\{ f \in L^2_{\mu}(X) :\, (\exists c \in \mathbb{R})\, f(x)=c , \, \mu-a.e. \right\}? $$
If so, how can be compute the optimizer?
Since $\ f^1, f^2, \dots, f^n\ $ and $\ y\ $ are all in $\ L_\mu^2(X)\ $, so is $\ y-\sum_\limits{i=1}^n\beta^i f^i\ $, and \begin{eqnarray} \int_{x \in X} (y(x)-\sum_{i=1}^n \beta^if^i(x))^2 d\mu(x)&=& \lVert y - \sum_{i=1}^n \beta^if^i\rVert^2\\ &=& \lVert y\rVert^2 + \sum_{i=1}^n\sum_{k=1}^n \langle f^i,f^j\rangle \beta^i\beta^j - 2\sum_{i=1}^nRe\langle y, f^i\rangle \beta^i\\ &=& \lVert y\rVert^2 + \beta^\top\,\Phi\beta - 2\psi^\top\beta\ , \end{eqnarray} where $\ \Phi\ $ is the $\ n\times n\ $ (real) matrix whose entry in its $\ i^\mathrm{th}\ $ row and $\ j^\mathrm{th}\ $ column is $\ Re\langle f^i,f^j\rangle\ $, $\ \psi\ $ the $\ n\times 1\ $ column vector whose $\ i^\mathrm{th}\ $ entry is $\ Re\langle y, f^i\rangle\ $, and $\ \beta\ $ the $\ n\times 1\ $ column vector whose $\ i^\mathrm{th}\ $ entry is $\ \beta^i\ $. The matrix $\ \Phi\ $ is symmetric, and will be positive definite whenever $\ f^i, i=1,2,\dots, n\ $ are linearly independent, or semi-definite otherwise. As a function of $\ \beta\ $, $\ \lVert y\rVert^2 + \beta^\top\,\Phi\beta - 2\psi^\top\beta\ $ is continuous and convex, and strictly convex whenever $\ \Phi\ $ is positive definite.
The solution to the problem of minimising a function of the form $\ \lVert y\rVert^2 + \beta^\top\,\Phi\beta - 2\psi^\top\beta\ $ is well known. If $\ \psi\in\mathcal{R}(\Phi)\ $, then a minimising value $\ \beta^*\ $ of $\ \beta\ $ is any solution of the equation $\ \Phi\beta^* = \psi\ $, because then $\ \lVert y\rVert^2 + \beta^\top\,\Phi\beta - 2\psi^\top\beta=$$\lVert y\rVert^2 + \left(\beta-\beta^*\right)^\top\,\Phi\left(\beta-\beta^*\right) - \beta^{*\top}\Phi\beta^*\ $. On the other hand, if $\ \psi\not\in\mathcal{R}(\Phi)\ $, then the function $\ \lVert y\rVert^2 + \beta^\top\,\Phi\beta - 2\psi^\top\beta\ $ is not bounded below, because there will exist a real vector $\ \alpha\in \mathcal{ker}(\Phi)\ $ with $ \langle\psi,\alpha\rangle > 0\ $, and the objective value can then be made arbitrarily small by choosing $\ \beta=c\alpha\ $ with $\ c\ $ arbitrarily large.