Page 10 - Vector Analysis

6 CHAPTER 1. Linear Algebra

1.3 Normed Vector Spaces

The norm introduced in Proposition 1.16 is a good way of measure the magnitude of vectors.
In general if a real-valued function can be used as a measurement of the magnitude of vectors
if certain properties are satisfied.

Definition 1.21. Let V be a vector space over scalar field F. A real-valued function } ¨ } :
V Ñ R is said to be a norm of V if

1. }v} ě 0 for all v P V.

2. }v} = 0 if and only if v = 0.

3. }αv} = |α|}v} for all v P V and α P F.

4. }v + w} ď }v} + }w} for all v, w P V.

The pair (V, } ¨ }) is called a normed vector space.

Example 1.22. Let V = Fn, and } ¨ }p be defined by

$[ n ]1
ÿ |xi|p p
’ if 1 ď p ă 8,
& i=1

}x}p =

’ max |xi| if p = 8 ,
%
1ďiďn

where x = (x1, ¨ ¨ ¨ , xn). The function } ¨ }p is a norm of Fn, and is called the p-norm of Fn.

Theorem 1.23 (Hölder’s inequality). Let 1 ď p ď 8. Then

ˇˇ(x, y)ˇˇ ď }x}p}y}p1 @ x, y P Fn , (1.1)

where (¨, ¨) is the standard inner product on Fn and p1 is the conjugate of p satisfying
1 1
p + p1 = 1.

Proof. Let x = (x1, ¨ ¨ ¨ , xn) and y = (y1, ¨ ¨ ¨ , yn) be given. Without loss of generality we

can assume that x ‰ 0 and y ‰ 0. Define x = x/}x}p and y = y/}y}p1 . Then }xr}p = 1 and
r r

}yr}p1 = 1. By Young’s inequality

ab ď 1 ap + 1 bp1 @ a, b ě 0 ,
p p1