The Chebyshev conjecture posits that Chebyshev subsets of a real Hilbert space $X$ are convex. Works by Asplund, Ficken and Klee have uncovered an equivalent formulation of the Chebyshev conjecture in terms of uniquely remotal subsets of $X$. In this tradition, we develop another equivalent formulation in terms of Chebyshev subsets of the unit sphere of $X\times \mathbb{R}$. We characterise such sets in terms of the image under stereographic projection. Such sets have superior structure to Chebyshev sets and uniquely remotal sets.

A Chebyshev set is a subset of a normed linear space that admits unique best approximations. In the first part of this paper we present some basic results concerning Chebyshev sets. In particular, we investigate properties of the metric projection map, sufficient conditions for a subset of a normed linear space to be a Chebyshev set, and sufficient conditions for a Chebyshev set to be convex. In the second half of the paper we present a construction of a nonconvex Chebyshev subset of an inner product space.

We find the potential function whose gradient best approximates an observed square integrable function on a bounded open set subject to prescribed weight factors. With an appropriate choice of topology, we show that the gradient operator is a bounded linear operator and that the desired potential function is obtained by solving a second-order, self-adjoint, linear, elliptic partial differential equation. The main result makes a precise analogy with a standard procedure for the best approximate solution of a system of linear algebraic equations. The use of bounded operators means that the definitive equation is expressed in terms of well-defined functions and that the error in a numerical solution can be calculated by direct substitution into this equation. The proposed method is illustrated with a hypothetical example.

We study the Friedrichs extensions of unbounded cyclic subnormals. The main result of the present paper is the identification of the Friedrichs extensions of certain cyclic subnormals with their closures. This generalizes as well as complements the main result obtained in [5]. Such characterizations lead to abstract Galerkin approximations, generalized wave equations, and bounded -functional calculi.

We combine the theory of sectorial sesquilinear forms with the theory of unbounded subnormal operators in Hilbert spaces to characterize the Friedrichs extensions of multiplication operators (with analytic symbols) in certain functional Hilbert spaces. Such characterizations lead to abstract Galerkin approximations and generalized wave equations.