Vol. 10 No 6 - Pi Mu Epsilon

478 PI Mu EPSILON JOURNAL
A VISUAL REPRESENTATION OF THE SEQUENCE SPACE
479
= Y" doo (tt- 1 (pj_ 1
) , tt- 1 (pj) )
/':'1
Since x- 1 (pj_ 1
) and x- 1 (pj) differ by only one digit in ~th place, (where ~
is the index of the j th term in which a differs from 't) we have
ct-cx- 1 (pj_ 1
), x- 1 (pj)) = 4- so
2 J
..,
.. ~
..
=~
= d..,(a,'t). 0
Thus, (P, d,) is a geometric model of the sequence space.
a u a
• • • •
2
(a.)
Figure 6
(b.)
p
•
I
- 2
u
•
Additional Theorems
We close by proving some of the claims stated in the introduction.
Theorem 4.1. There does not exist an isometric embedding from
(E , d..,) into (Jln, ds> where d• denotes the Euclidean metric,
ds(x,y) = lx- yj.
Proof. Since (1: , dz) can be considered to be a subspace of
(!:,d .. ) it suffices to show that 1: does not embed isometrically into
lln. Suppose k : Lz - Jln is an isometric bijection. Then
dz (u, 13> = jk(u) - k I for any u, 13 E 1: . Consider the points:
a = k(OO)
't = 1t(01)
p = x(10)
v = k(ll)
Ia - -rl = d 2 (00, 01)
lor- vi= dz(01,ll)
Ia- vi= dz(OO,ll)
1
- 2
= 1
3
- 2
Thus a,