# IJPAM: Volume 5, No. 4 (2003)

**ALPHA-DENSE CURVES IN INFINITE**

DIMENSIONAL SPACES

DIMENSIONAL SPACES

Department of Mathematical Analysis

Faculty of Sciences

University of Alicante

Ap. Correus 99, E-03080 Alicante, SPAIN

e-mail: gaspar.mora@ua.es

e-mail: jamira@ua.es

**Abstract.**The theory of dense curves in the euclidean space
was developed for finding algorithms for Global Optimization of
multivariable functions (
,
). The -dense curves, considered as a generalization of Peano curves or
space-filling curves, densify the domain of definition of a
multivariable function in the sense of the Hausdorff metric. Then, the
restriction of on an dense curve , contained in ,
is a univariable function for which will have less difficulty
to locate its global minimum.

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

Finally, we shall construct a family of infinite dimensional dense
curves, whith controlled density , 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