Lecture Notes in Differential Equations - Bruce E. Shapiro

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Example 15.3. By a similar argument as in the previous problem, the set
of all functions
f(t) : R n → R m (15.15)
is also a vector space, using the usual definitions of function addition and
multiplication by a constant.
Definition 15.2. Let V be a vector space. Then a norm on V is any
function ‖y‖ : V → R (i.e., it maps every vector y in V to a real number
called ‖y‖) such that
1. ‖y‖ ≥ 0 and ‖y‖ = 0 ⇐⇒ y = 0.
2. ‖cy‖ = |c|‖y‖ for any real number c, vector y.
3. ‖y + z‖ ≤ ‖y‖ + ‖z‖ for any vectors y, z. (Triangle Inequality)
A normed vector space is a vector space with a norm defined on it.
Example 15.4. In the usual Euclidean vector space, the 2-norm, given by
‖v‖ = √ x 2 + y 2 + z 2 (15.16)
where the positive square root is used. You probably used this norm in
Math 280.
Example 15.5. Another norm that also works in Euclidean space is called
the sup-norm, defined by
Checking each of the three properties:
‖v‖ ∞ = max(|x|, |y|, |z|) (15.17)
1. ‖v‖ ∞ is an absolute value, so it cannot be negative. It can only be
zero if each of the three components x = y = z = 0, in which case
v = (0, 0, 0) is the zero vector.
2. This follows because
‖cv‖ ∞ = ‖c(x, y, z)‖ ∞ (15.18)
= ‖(cx, cy, cz))‖ ∞ (15.19)
= max(|cx|, |cy|, |cz|) (15.20)
= |c| max(|x|, |y|, |z|) (15.21)
= |c|‖v‖ ∞ (15.22)