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2.5 Vector Spaces, Norms, and Inner Products 39

EXAMPLE 2.4–5: Equilibrium Points of Pendulum

The equilibrium points of the pendulum are isolated. On the other hand, a
system described by =0 has for equilibrium points any point in R and
therefore, none of its equilibrium points is isolated.

2.5 Vector Spaces, Norms, and Inner Products

In this section, we will discuss some properties of nonlinear differential
equations and their solutions. We will need many concepts such as vector
spaces and norms which we will introduce briefly. The reader is referred to
[Boyd and Barratt], [Desoer and Vidyasagar 1975], [Khalil 2001] for proofs
and details.

Linear Vector Spaces
In most of our applications, we need to deal with (linear) real and complex
vector spaces which are defined subsequently.

DEFINITION 2.5–1 A real linear vector space (resp. complex linear vector
space is a set V, equipped with 2 binary operations: the addition (+) and the
scalar multiplication (.) such that

Copyright © 2004 by Marcel Dekker, Inc.