“The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena.”

David Hilbert

1.1 Introduction

The basic algebraic concepts in the theory of Hilbert spaces are those of a vector space and an inner product. The inner product defines a norm, and thus every Hilbert space is a normed space. Since the norm plays a very important role in the theory, it is not possible to study Hilbert spaces without familiarity with basic concepts and properties of normed spaces. Section 1.2 provides a brief discussion of the algebraic structure of vector spaces. Section 1.3 discusses topological aspects of normed spaces. Basic properties of complete normed spaces are presented in Section 1.4. The final two sections of this chapter are devoted to linear mappings in normed spaces and the fixed point theorem.

This chapter is by no means a substitute for a course in the theory of normed spaces. We limit our discussion to those concepts which are necessary for understanding of the following chapters. Some of the definitions and theorems do not require the algebraic structure of vector spaces and can be formulated in metric spaces or general topological spaces. We choose not to make these distinctions and always assume that we are dealing with a vector space or a normed vector space.

1.2 Vector Spaces

We consider both real and complex vector spaces. The field of real numbers is denoted by and the field of complex numbers by . Elements of or are called scalars. Sometimes it is convenient to give a definition or state a theorem without specifying the field of scalars. In such a case we use to denote either or . For instance, if is used in a theorem, this means that the theorem is true for both scalar fields and .

From (c) it follows that for every x ∈ E there exists a zx ∈ E such that x + zx = x. We will show that there exists exactly one element z ∈ E such that x + z = x for all x ∈ E. That element is denoted by 0 and called the zero vector.

Let x, y ∈ E. By (c), there exists a w ∈ E such that x + w = y. If x + zx = x for some zx ∈ E, then, by (a) and (b),

This shows that, if x + z = x for some x ∈ E, then y + z = y for any other element y ∈ E. We still need to show that such an element is unique. Indeed, if z1 and z2 are two such elements, then z1 + z2 = z1 and z1 + z2 = z2. Thus, z1 = z2.

A similar argument shows that the vector z in (c) is unique for any pair of vectors x, y ∈ E.

The unique solution z of x + z = y is denoted by y − x. According to the definition of the zero vector, we have x − x = 0. The vector 0 − x is denoted by -x.

We use 0 to denote both the scalar 0 and the zero vector; this will not cause any confusion. Because of (b), the use of parentheses in expressions with more than one plus sign can be avoided.

The following properties follow easily from the definition of vector spaces:

A subset E1 of a vector space E is called a vector subspace (or simply a subspace if for every α, β ∈ and x, y ∈ E1 the vector αx + βy is in E1). Note that a subspace of a vector space is a vector space itself. According to the definition, a vector space is a subspace of itself. If we want to exclude this case, we say a proper subspace, that is, E1 is a proper subspace of E if E1 is a subspace of E and E1 ≠ E.

If E1 and E2 are subspaces of a vector space E and E1 ⊆ E2, then E1 is a subspace of E2. For instance, the space of all polynomials of N variables is a subspace of C∞(N), which in turn is a subspace of Ck(N) or C(N). The intersection of subspaces of a vector space is always a subspace, but the same is not true for the union of subspaces.

We use the notation (xn) or (x1, x2, …) to denote a sequence whose nth term is xn, and {xn: n ∈ } to denote the set of all elements of the sequence. Note that {xn: n ∈ } can be a finite set even though (xn) is an infinite sequence.

In most examples, verifying that a set is a vector space is easy or trivial. In...