IJPAM: Volume 5, No. 4 (2003)

ALPHA-DENSE CURVES IN INFINITE
DIMENSIONAL SPACES

Gaspar Mora$^1$, Juan A. Mira$^2$
Department of Mathematical Analysis
Faculty of Sciences
University of Alicante
Ap. Correus 99, E-03080 Alicante, SPAIN
$^1$e-mail: gaspar.mora@ua.es
$^2$e-mail: jamira@ua.es


Abstract.The theory of $\alpha -$dense curves in the euclidean space $R^{n}, n\geq
2, $ was developed for finding algorithms for Global Optimization of multivariable functions ( $\left[ 1\right]$, $\left[ 6\right]$). The $%
\alpha $-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 $\alpha -$dense curve $\gamma $, contained in $D$, is a univariable function $f_{\gamma }$ for which will have less difficulty to locate its global minimum.

In this paper we shall study some properties of $\alpha -$dense curves that are Lipschitzian. Moreover, we shall point out that this theory of $%
\alpha -$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 $\alpha. $ From this, we shall deduce the classical Theorem of Riesz.

Finally, we shall construct a family of infinite dimensional $\alpha -$dense curves, whith controlled density $\alpha $, in the Hilbert parallelotope.

Received: January 10, 2003

AMS Subject Classification: 46B25, 14H50, 28A12

Key Words and Phrases: alpha-dense curves, space-filling curves, functional analysis

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2003
Volume: 5
Issue: 4