Approximative properties of proximal subspaces of infinite dimension

For subspaces $L$ of infinite dimension in a Banach space, the authors obtained the characteristic properties of the existence of elements of the best approximation. As an application, they prove that, in the space $C(T)$ of continuous functions on a connected Hausdorff compactum $T$, the Chebyshev subspace $L\subset C(T)$ of infinite dimension, the annihilator $L^\perp$ of which is separable and contains the minimal total subspace, is a hyperplane $L=\mathrm{ker}(\alpha)$ of a strictly positive functional $\alpha\in L^\perp$.