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5.1 Vector Norms 275

To see that lim p→∞ x p = max i |x i | , proceed as follows. Relabel the en-
tries of x by setting ˜x 1 = max i |x i | , and if there are other entries with this
same maximal magnitude, label them ˜x 2 ,..., ˜x k . Label any remaining coordi-
nates as ˜x k+1 ··· ˜x n . Consequently, |˜x i /˜x 1 | < 1 for i = k +1,...,n, so, as
p →∞,
1/p
n p 1/p
p
p ˜x k+1 ˜x n
x = |˜x i | = |˜x 1 | k + + ··· + →|˜x 1 | .
p
˜ x 1 ˜ x 1
i=1
Example 5.1.2
To get a feel for the 1-, 2-, and ∞-norms, it helps to know the shapes and relative
sizes of the unit p-spheres S p = {x | x =1} for p =1, 2, ∞. As illustrated
p
3
in Figure 5.1.4, the unit 1-, 2-, and ∞-spheres in are an octahedron, a ball,
and a cube, respectively, and it’s visually evident that S 1 fits inside S 2 , which
3
in turn fits inside S ∞ . This means that x 1 ≥ x 2 ≥ x ∞ for all x ∈ .
n
In general, this is true in (Exercise 5.1.8).

S 1 S 2 S∞
Figure 5.1.4
Because the p-norms are defined in terms of coordinates, their use is limited
to coordinate spaces. But it’s desirable to have a general notion of norm that
works for all vector spaces. In other words, we need a coordinate-free definition
of norm that includes the standard p-norms as a special case. Since all of the p-
norms satisfy the properties (5.1.7), it’s natural to use these properties to extend
the concept of norm to general vector spaces.

General Vector Norms

A norm for a real or complex vector space V is a function mapping
V into that satisfies the following conditions.

x ≥ 0 and x =0 ⇐⇒ x = 0,
αx = |α| x for all scalars α, (5.1.9)
x + y ≤ x + y .