Show that x1 is a vector norm by verifying A function n is a norm provided: 1. for all if and only if (positivity); 2. for all (scaling); 3. for all n (triangle inequality). In this book,...

Show that x1 is a vector norm by verifying

A function

ⁿ

is a norm provided:

1.

for all

if and only if

(positivity);

2.

for all

(scaling);

3.

for all

ⁿ
(triangle inequality).

In this book, the only vector norms we will use are the
-norms defined by

for
The values

and

are the most commonly used norms. Equation 7.1 corresponds to

and we will refer to it as the Euclidean norm or the 2-norm, and indicate this using the notation

For

and

the norms are

Problem 7.9 justifies the formula for