Alpha-dense curves in infinite dimensional spaces

Abstract

The theory of α−dense curves in the euclidean space Rn , n ≥ 2, was developed for finding algorithms for Global Optimization of multivariable functions ([1], [6]). The α-dense curves, considered as a generalization of Peano curves or space-filling curves, densify the domain of definition D of a multivariable function f in the sense of the Hausdorff metric. Then, the restriction of f on an α−dense curve γ, contained in D, is a univariable function fγ for which will have less difficulty to locate its global minimum. In this paper we shall study some properties of α−dense curves that are Lipschitzian. Moreover, we shall point out that this theory of α−dense curves is characteristic of the finite dimensional spaces. In fact, we shall prove that a Banach space has finite dimension iff its unit ball can be densified with arbitrary small density α. From this, we shall deduce the classical Theorem of Riesz. Finally, we shall construct a family of infinite dimensional α−dense curves, whith controlled density α, in the Hilbert parallelotope.