Graphics Reference
In-Depth Information
2
5,
--
(
XY
XYZ
cos d
=
.
)
,
If
p
corresponds to a real point (x,y), that is,
p
= [x,y,1], the only way that d will go
to zero is if x and y both get arbitrarily large and (x,y) is near the line y - 2x = 0. This
follows from the Cauchy-Schwarz inequality. Similarly, the only ideal points close to
L
are points
p
= [X,Y,0] with X close to -2 and Y close to -1.
N-dimensional projective space
P
n
5.9.2. Theorem.
is a compact, connected,
metrizable topological manifold.
Proof.
The compactness and connectedness follows from Lemma 5.3.18 and Theo-
rems 5.4.6 and 5.5.2 using the third and fourth definition of
P
n
. We postpone showing
that
P
n
is a manifold to Section 8.13, where we will in fact show that it is a differen-
tiable manifold.
Finally, we note that any hyperplane in
P
n
is homeomorphic to
P
n-1
. In particu-
lar, the subspace of ideal points is homeomorphic to
P
n-1
.
5.10
E
XERCISES
Section 5.2
5.2.1.
Prove that every metric on a finite set is the discrete metric.
5.2.2.
Prove that if a vector space
V
has an inner product <,>, then the function
(
)
==<-->
d
uv uv
,
vuvu
,
defines a metric on
V
.
Prove that equation (5.1) defines an inner product on C
0
([0,1]).
5.2.3.
Prove that the function d
1
defined by equation (5.3) defines a metric on C
0
([0,1]).
5.2.4.
Prove that the function d
defined by equation (5.4) defines a metric on C
0
([0,1]).
5.2.5.
Show that the metrics d
1
and d
on C
0
([0,1]) defined by equations (5.3) and (5.4), respec-
tively, are not equivalent metrics.
5.2.6.
5.2.7.
Let (
X
,d) be a metric space. Prove that the function d* defined by equation (5.5) is a
bounded metric on
X
.
Consider the sequence of functions f
n
(x) = x
n
on [0,1]. Show that this sequence of func-
tions converges in a pointwise fashion but not uniformly. Note also that although each
function is continuous, the limit function g(x) is not. Describe g(x).
5.2.8.
5.2.9. Show that the rational numbers are not complete by giving an example of a Cauchy
sequence of rational numbers that does not converge to a rational number.
5.2.10. Sometimes one does not quite have a metric on a space.
