3.1 Introduction

In this chapter, key definitions, theorems, and techniques from linear algebra and functional analysis are briefly reviewed through examples. The purpose is to introduce notations and concepts that will recur in subsequent chapters.

In Section 3.2, definitions of vector space, norm, and inner product are given along with examples. In Section 3.3, linear dependence and independence of vector valued functions and certain properties of time-varying matrices are discussed. Section 3.4 contains definitions and examples of bounded linear operators, Banach spaces, Hilbert spaces, and adjoint operator. In Section 3.5, the Banach fixed point theorem and some examples are given. In Section 3.6, compact set, compact operator, closed operator, and open mapping are discussed briefly.

Section 3.6 is included primarily to support Chapter 10 on infinite dimensional systems. Readers without a background in basic functional analysis may skip both Section 3.6 and Chapter 10.

3.2 Vector Space

Definition 3.2.1 A field 𝔽 is a set of elements, called scalars, together with two operations, addition and multiplication, for which the following axioms hold:

• (i) For a,b ∈ 𝔽, a+b ∈ 𝔽 and ab ∈ 𝔽. Moreover, a+b = b+a and ab = ba (the addition and multiplication are commutative).

• (ii) For a,b,c ∈ 𝔽, a + (b + c) = (a + b) + c and a(bc) = (ab)c (the addition and multiplication are associative).