Chemistry Reference
In-Depth Information
implying that
k
v
kk
w
kk
v
w
k
and
k
w
kk
v
kk
v
w
k
:
Therefore
ˇ
ˇ
k
v
kk
w
k
ˇ
ˇ
k
v
w
k
;
showing that the distance between
k
v
k
and
k
w
k
cannot exceed the distance between
v
and
w
.
We can also show that with any vector norm the circular neighborhoods of any
vector
u
2 R
n
, denoted
n
;
k
v
u
k
<"
g
;
U
Df
v
j
v
2 R
are convex sets. In order to prove this property assume that
x
and
y
are in U ,then
both
k
x
u
k
and
k
y
u
k
are less than ". With any vector
z
D
˛
x
C
.1
˛/
y
.0
˛
1/;
we have
k
z
u
kDk
˛.
x
u
/
C
.1
˛/.
y
u
/
k
j
˛
jk
x
u
kCj
1
˛
jk
y
u
k
<˛"
C
.1
˛/"
D
";
which proves the assertion.
Consider a linear segment
u
C
t.
v
u
/.0
t
1/ connecting points
u
and
v
in
R
n
.Let0
D
t
0
<t
1
<
<t
k
D
1 and
u
l
D
u
C
t
l
.
v
u
/ for l
D
0;1;:::;k.
Then
X
lD1
k
u
l
k
k
v
u
kD
u
l1
k
;
(A.3)
since
k
k
X
X
1
k
u
l
u
l1
kD
1
k
.
u
C
t
l
.
v
u
//
.
u
C
t
l1
.
v
u
//
k
l
D
l
D
X
k
X
k
D
1
k
.t
l
t
l1
/.
v
u
/
kD
.t
l
t
l1
/
k
v
u
k
l
D
l
D
1
D
.t
k
t
0
/
k
v
u
kDk
v
u
k
:
n
is a continuous function. We can easily prove
Assume next that
f
W
Œa;b
7! R
that
Z
b
Z
b
f
.t/dt
f
.t/
dt:
(A.4)
a
a
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