May 2024
Intermediate to advanced
687 pages
18h 35m
English
Content preview from Discrete Mathematical Structures by Pearson
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12-48 Discrete Mathematical Structures
Example 4 On the space C[a, b], norm is defined by
ff
xd
x
a
b
=
∫
|()| .
Verify norm axioms.
Solution:
(1) If f (x) = 0 for all x ∈C[a, b], then
ffxd
xd
x
a
b
a
b
==
=
∫∫
|()| .
00
When
f
≠
0 , then |f (x)| is positive for all x in [a, b] or in some subinterval of [a, b] and
therefore
ffxdx
a
b
=>
∫
|()| 0
(2) Since |f (x) + g(x)| ≤ |f (x)| + |g(x)| for every x,
|()()| |()| |()|fx gx dx fx dx gx
dx
a
b
a
b
a
b
+≤ +
∫∫
∫
i.e.,
fg
fg
+≤ +
(3)
αα
αα
ffxdxfxd
xF
a
b
a
b
== ∈
∫∫
|()| |()| ,
= |a| ||
f ||
Hence, the norm axioms are verified.
12.8.2 Inner Product
The angle between two non-zero vectors in either R
2
or R
3
can be defined by using the concept ...
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