Vol. 10 No 6 - Pi Mu Epsilon

466 PI Mu EPSILON JOURNAL
ViSUAL REPRESENTATION OF THE SEQUENCE SPACE
467
Acknowledgment
I am grateful to Dr. Prem N. Bajaj who is an Associate Professor in the
Mathematics and Statistics Department in The Wichita State University. I
am thankful for his help and advice during my paper preparations. Dr.
Bajaj has been very patient with me and he encourages me in many ways.
I am also grateful to the referee's valuable comments on my paper.
Reference
1. Morris W. Hirsch and Stephen Smale, Differential Equations,
Dynamical Systems, And Linear Algebra, Academic Press, New York,
1974.
Gee Yoke Lao and her husband came to The Wichita State University
from Singapore in 1993. In May of 1996, they both received a bachelor's
degree in Electrical Engineering. Gee is working on a master's degree in
Electrical Engineering at WSU.
A VISUAL REPRESENTATION OF
THE SEQUENCE SPACE
Marc Fusaro (student)
University of Scranton
Introduction
In an undergraduate course on chaotic dynamical systems based on
Devaney's text [1], one encounters a strange metric space called the
sequence space, CE , doo). This is a very difficult space to visualize. In
fact, it can easily be shown that this space cannot be embedded isometrically
in Rn for any n. In this paper we develop a geometric model of L which
gives a very good impression of the geometric structure of the space. The
model consists of a subset of the Euclidean plane and an unusual, but easy
visualize, metric on these points making the model isometric to CE . doo).
Definition 1.1. The sequence space on two symbols is the set
L = {s 0 s 1 SzS 3 ••• I s 1 E {011}}
• the distance between two elements of this set defined by
d..,(a ~) = ~ lsi- til
I ~ 2j
a = s 0
s 1
Sz. • • and ~ = t 0 t 1 t 2
• • • • That is, L is the set of all
sequences of zeros and ones. . . .
While the points and distances are clearly defined, 1t IS very dtfficult to
ize the geometric structure of the space as a whole. The goal of this
s-P!r is to construct a geometric model of the sequence space as an aide to
visualizing its geometry.
In order to gain some insight into the geometry of E we begin by
considering a family of finite subspaces of I: .
Definition 1.2. Define the family of metric spaces (En 1 dn) by
En = {s 0
s 1
s 2
••• sn- 1 1 s 1 E {0~1}},
n-1 lsi- tll
dn (a 1 ~) = ~ j )
~ 2
;here a = s 0
• •• sn-l and ~ = t 0
• •• tn_ 1
• That is En , is what we get if we
consider only the first n terms of an element of E .
Clearly doo and dn are metrics and (En 1 dn) and (E 1
doo) are metric
spaces . . .
Perhaps the simplest model one could make for a gtven metnc space
v.'Ould be to embed it isometrically into some Euclidean space (preferably
or R2 or R3). We will show in Section Four that even the tiny subspace Ez
of L can not be isometrically embedded in R 0 , no matter bow large we
dloose n. Thus R 0 with the Euclidean metric fails to contain an embedded
model for L. .
However, if we switch to the grid metric (often called the taxtcab
metric) we can construct a model of En in R".
Definition 1.3. Consider R 0 with the grid metric
:.