Page 12 - Vector Analysis

8 CHAPTER 1. Linear Algebra

Proof. By Hölder’s inequality, it is clear that }x}p ě sup ˇˇ(x, y)ˇˇ for all x P Fn. On the

}y}p1 =1

other hand, note that |xk|p = xk ¨xk|xk|p´2; thus letting yk = xk |xk |p´2 we find that }y}p1 = 1
}x}pp´1
which implies that
n
ˇˇ(x, y)ˇˇ = 1 = }x}p
}x}pp´1 ÿ |xk|p

k=1

which implies that sup ˇˇ(x, y)ˇˇ ě }x}p. ˝

}y}p1 =1

Making use of Hölder’s inequality (1.1) and the Riemann sum approximation of the

Riemann integral, we can conclude the following

Theorem 1.26. Let 1 ď p ď 8. If p1 is the conjugate of p; that is, 1 + 1 = 1, then
p p1

ˇż1 ˇ

ˇ f (x)g(x) dxˇ ď }f }p}g}p1 @ f, g P C ([0, 1]; R) ,
ˇ ˇ

0

where $ (ż1 )1

’ |f (x)|pdx p if 1 ď p ă 8 ,
’ if p = 8 .
&

}f }p = 0

’ max |f (x)|
’

% xP[0,1]

Remark 1.27. The Minkowski inequality implies that

}f + g}p ď }f }p + }g}p @ f, g P C ([0, 1]; R) .

In other words, the function } ¨ }p : C ([0, 1]; R) Ñ R is a norm on C ([0, 1]; R), and is called
the Lp-norm.

1.4 Matrices

Definition 1.28 (Matrix). Let F be a scalar field. The space M(m, n; F) is the collection

of elements, called an m-by-n matrix or m ˆ n matrix over F, of the form


a11 a12 ¨ ¨ ¨ a1n
A =   ,
a21 a22 ¨¨¨ a2n
... ... ... ...

am1 am2 ¨ ¨ ¨ amn