7.1 Banach Algebras

In a first course in linear algebra, one encounters a variety of interesting ideas surrounding linear operators on finite dimensional spaces. The central object of this discussion is the concept of an eigenvalue or characteristic value. The eigenvalues of an operator togetherwith their eigenvectors help us build a collection of ideas such as the Cayley–Hamilton Theorem, the question of diagonalizability of an operator, the theory of canonical forms and much much more.

In this chapter, we will revisit the idea of an eigenvalue in the context of operators on infinite dimensional spaces. Here, the set of eigenvalues needs to be replaced by the spectrum of an operator. This is a compact subset of the complex plane that carries a great deal of information about the operator and is the object we wish to study.

Note. Before we get going, wemake one important assumption.Henceforth all vector spaces will be over C. The precise reason for this will be explained in due course, but it is related to the Fundamental Theorem of Algebra.

This chapter examines the importance of natural gas for electricity generation in Australia. Gas-fired power plants use either a boiler to create steam, which turns the turbine and generates electricity, or a combustion turbine to create a rotating mass that generates electricity. Natural gas is also used commercially and domestically for cooking and heating purposes. In both instances, once extracted, natural gas must be transported from the well-head through gathering and processing facilities into storage or transportation pipelines. Gas has become one of the primary fossil fuels used to generate electricity in Australia. Electricity is aggregated and sold through the national energy market. This is set up by the National Electricity Law (NEL). The transportation and distribution of gas is regulated by the National Gas Law. The retail sale of electricity is regulated by the National Electricity Retail Law. This chapter examines how the national energy market has been established under the NEL.

Risk is lack of information about the future. A situation is risky if it has widely varying possible outcomes and there’s no way to determine with high confidence which outcome will occur. A riskless or risk-free situation is one whose future is known exactly.

6.1 Weak Convergence

In Chapter 4, we saw a number of interesting ways in which a normed linear space interacts with its dual space. Notably, the Hahn–Banach Theorem gave us a powerful way to construct bounded linear functionals. In this chapter, we explore this relationship further and once again the Hahn–Banach Theorem plays a central role.

We will begin by exploring the notion of weak convergence. The ideas developed in this section will, together with a geometric version of the Hahn–Banach Theorem, help us define new topologies on a normed linear space and its dual space. These topologies are weaker (andmore forgiving) than the norm topology, which allows us to prove some powerful compactness theorems.

Benjamin Graham famously anthropomorphized the US stock market, attributing wild emotional swings to “Mr. Market.” Sometimes Mr. Market was fearful, usually after a sufficiently traumatic negative event.

This chapter explores cultural environments, including a look at the nature and characteristics of cultures and subcultures; several contemporary models of national cultures; recent refinements to the models and how they can help managers understand some of the nuances of cultural differences; how managers can use these models to better understand managerial and organizational behavior; and the challenges and opportunities of cultural diversity and multiculturalism.

1.1 Review of Linear Algebra

Linear algebra is primarily the study of finite dimensional vector spaces and linear transformations between them. In a first course, one encounters the fact that every such vector space has a basis, that linear transformations may be associated to matrices, and one finally understands why matrix multiplication looks so clumsy. This section sets out to relive those glory days.

1.1.1 DEFINITION A vector space over a field K is a set E together with two operations a: E × E → E (vector addition) and s: K × E → E (scalar multiplication) written as a(x, y) = x + y and s(α, x) = αx, which satisfy the following properties.

In the previous chapter, we saw a variety of distributions that can be used for situations that fall into Knight’s a priori risk category. If we are confident that a known distribution describes all the outcomes and all the associated probabilities for a set of variables, then we might even be able to get a closed-form description of relevant risk metrics for these variables.

Chapter 3 investigated the seeming impossibility of uncertainty arising from certainty: even when we know exactly when and how much money we will get in the future, the value we assign today to that future money can be highly variable.