Lecture Notes in Differential Equations - Bruce E. Shapiro

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Therefore
‖y + z‖ 2 ≤ ‖y‖ 2 + 2‖y‖‖z‖ + ‖z‖ 2 = (‖y‖ + ‖z‖) 2 (15.34)
Taking square roots of both sides gives the third property of norms.
Example 15.7. Let V be the vector space consisting of integrable functions
on an interval (a, b), and let f ∈ V. Then the sup-norm defined by
is a norm.
‖f‖ ∞ = sup{|f(t)| : t ∈ (a, b)} (15.35)
The first property follows because it is an absolute value. The only way
the supremum of a non-negative function can be zero is if the function is
identically zero.
The second property follows because sup |cf(t)| = |c| sup |f(t)|
The third property follows from the triangle inequality for real numbers:
Hence
|f(t) + g(t)| ≤ |f(t)| + |f(t)| (15.36)
‖f + g‖ = sup |f(t)+g(t)| ≤ sup |f(t)|+sup |f(t)| = ‖f‖+‖g‖ (15.37)
Definition 15.3. A Linear Operator is a function L : V ↦→ V whose domain
and range are both the same vector space, and which has the following
properties:
1. Additivity. For all vectors y, z ∈ V,
L(y + z) = L(y) + L(z) (15.38)
2. Homogeneity. For all numbers a and for all vectors y ∈ V,
These two properties are sometimes written as
L(ay) = aL(y) (15.39)
L(ay + bz) = aL(y) + bL(z) (15.40)
It is common practice to omit the parenthesis when discussing linear operators.