2.1 Definitions and Examples

We now introduce the fundamental object in our investigations. A vector space, while very useful, is somewhat unwieldy when it is infinite dimensional. Equipping it with a metric, especially one that understands the linear structure of the underlying set, is a simple and effective way to alleviate this problem.

2.1.1 DEFINITION A norm on a K-vector space E is a function

which satisfies the following properties for all x, y ∊ E and all α ∊ K.

In the previous chapter, we saw that finding a Markowitz efficient frontier in an equality-constrained setting was simple: just use Lagrange multipliers and specify a closed-form solution. But finding an efficient frontier got more complicated in Section 4.1.3 when the constraints were inequalities. The example we used there with only three assets was simple enough to think through explicitly.

In mathematical terms, an operator transforms a given function into a new function. To better understand operators, we summarise the analogies with matrices below.

9.1 Positive Linear Functionals on C*-Algebras

In our quest to understand operators on a Hilbert space, we now take a detour. The purpose of this chapter is to prove a deep relationship between measures and linear functionals. The main theoremin this context is the Riesz–Markov–Kakutani Theorem, one of the most important theorems in modern analysis (this theorem is also referred to as the Riesz Representation Theorem, but we choose to use the longer name so as to prevent any confusion with Theorem 3.2.2). We will then use this result to identify the dual space of C(X), the space of continuous functions on a compact metric space X.

We begin by studying positivity in the context of C*-algebras; a notion that was baked-in to the theory by the founding fathers of the subject.

9.1.1 DEFINITION An element a in a C*-algebra A is said to be positive if it is self-adjoint and σ(a) ᑕ [0,∞). Wewrite A+ for the set of all positive elements in A and write a ≥ 0 if a ∊ A+.

The next lemma tells us where to look for positive elements and allows us to construct some simple examples.

9.1.2 LEMMA If a ∊ A is positive, then there exists b ∊ A such that b*b = a.

Proof. Suppose first that A is unital and let f ⟼ f (a) denote the continuous functional calculus. Since σ(a) ᑕ [0,∞), g(t) := t1/2 defines a continuous function on σ(a) and b := g(a) satisfies the required condition.

If A is non-unital, then we may apply the same argument in the unitalization Au. Since g(0) = 0, Remark 8.5.4 tells us that b ∊ A.

Australia has a vast range of renewable and non-renewable energy resources. These resources generate energy for domestic and international consumers for a range of different residential and industrial purposes. The acceleration of climate change and the need to reduce anthropogenic greenhouse gas emissions has opened up new opportunities to generate energy in a less carbon-intensive manner. The shift away from carbon-intensive fossil fuel energy generation has accelerated markets for renewable energy generation from kinetic processes like solar, wind and hydrogen. Historically, the energy framework in Australia has been dominated by non-renewable energy generation. This is largely a consequence of the fact that Australia has extensive coal and gas reserves. Black and brown coal reserves are particularly prevalent in the eastern states of New South Wales and Victoria. Australia’s identified conventional gas resources are extensive and extraction has increased threefold over the past two decades despite the accelerating climate emergency. Most of the recoverable reserves of conventional gas are located off the west and north-west coasts.

This chapter raises two simple yet important questions. First, what have we learned in our exploration of global management? And second, where do we go from here?

This chapter explores topics related to key questions about leadership in global settings, including: the meaning of leadership; how Eastern and Western leadership traditions differ; the meaning of global leadership; a look at two leadership models; the role of gender in leader behavior and success; and evaluating global leadership outcomes and effectiveness

In the previous chapter, we saw that markets in effect have moods – at times nervous, at times overconfident – that persist for a while but eventually revert to some long-term middle-of-the-road mood. We don’t find it unusual when our high-strung friend has a meltdown because his socks don’t match, but the same behavior in someone who is normally unflappable makes us sit up and take notice. It’s the sudden change from a calm mood to panic, and the eventual relaxation back to calm, that produces fat-tailed distributions.

Chapter 6 presents an overview of the organization of the mental grammar. We will focus on general architectural properties of the mental grammar, that is, the units and rules that every grammar must have to capture the sound form, meaning, and syntactic structure of words and sentences. I will suggest that the grammar functions like a checking device in that it tells the language user whether linguistic expressions are well-formed (i.e., grammatical, in accordance with the rules of grammar). There is some technical detail (and many linguistic terms), but at the very least the reader will be left with the conviction that languages are quite complex. It is explained how languages allow people to express any thought they might have, drawing attention to the pivotal notion of recursivity. This chapter sets the stage for being amazed that children have pretty much full control of their language by the age of 4. By learning what a mental grammar might look like, the reader can form an idea of what it is that the child needs to acquire. Without such information, it would be difficult to discuss the role of nature and nurture in language.

This chapter explores what is involved in global assignments, from moving abroad to returning home. Topics include: the purposes and types of global assignments; potential benefits and challenges of living and working globally; dealing with culture shock; choosing an acculturation strategy; and successful repatriation

3.1 Orthogonality

Hilbert space theory is a blend of Euclidean geometry and modern analysis. The inner product between vectors affords us the notion of orthogonality. This gives us access to geometric ideas such as Pythagoras’ Theorem (yes, really) and a best approximation property which formalizes the idea of ‘dropping a perpendicular’. This property, together with the notion of an orthonormal basis, will lead us to discover the Fourier series of an L2 function. These ideas were originally studied in the context of integral equations and are the historical roots of all of Functional Analysis.

David Hilbert (1862–1943) is perhaps best known for the twenty-three problems he posed in the International Congress of Mathematicians (ICM) in Paris in 1900. These problems speak to the incredible range of knowledge that Hilbert possessed and have been important signposts for mathematics in the 20th century. The Paris address was sandwiched between his immense work on axiomatizing geometry (published in 1899) and hiswork on integral equations (1904–1910) which laid the foundations for modern analysis. His mentorship of extraordinary students, including Hermann Weyl, Ernst Zermelo, Max Dehn, and many others, has only enriched his mathematical legacy.

We begin with some definitions and notation. Throughout this section, we will use the letter H to denote a Hilbert space and 〈·, ·〉 to denote the inner product on H.