We introduce the dual space of a Banach space X, which is the collection of all bounded linear maps from X into the field K (‘linear functionals’), equipped with the corresponding operator norm. We prove the Riesz Representation Theorem, which shows that in a Hilbert space, any linear functional can be written as the inner product with some element of the Hilbert space.

This chapter is dedicated to reviewing some of the basic concepts in RF design such as available power gain, matching circuits, and scattering parameters. We also present a more detailed discussion on both lossless and low-loss transmission lines and introduce the Smith chart. In addition, we recast a follow-up discussion on a receive–transmit antenna pair viewed as a two-port system. Most of the material presented will be used in Chapters 5 and 7, when we discuss noise and low-noise amplifiers.

We define what it means for a linear operator to be compact and show that the space of all compact linear operators is complete (with the operator norm). We give some examples and show that the spectrum of a compact operator from an infinite-dimensional space into itself is always non-empty (it must contain zero).

In this chapter we present a detailed discussion on various types of oscillators, including ring and crystal oscillators. The LC resonators and integrated capacitors and inductors have been already discussed in Chapter 1, and are essential to this chapter. Furthermore, some of the communication concepts that we presented in Chapter 2, such as AM and FM signals, as well as stochastic processes, are frequently used in this chapter.

The appropriate method of data analysis depends upon a variety of factors that have been specified in the research question and as part of the research design. One key issue is whether the data are qualitative or quantitative, and this depends upon the underlying research approach. If the research approach is deductive, then most of the data are likely to be expressed as numbers and the key issue will be selecting the appropriate statistical techniques for describing and analysing the data. In this chapter, we will concentrate on techniques for describing quantitative data and for providing simple preliminary analyses.

We recall some of the basic theory of linear algebra, beginning with the formal definition of a vector space. We then discuss linear maps between vector spaces and end by proving that every vector space has a basis using Zorn’s Lemma.

We define weak convergence and give some examples, including a proof of Schur’s Theorem that weak and strong convergence coincide in l^1. We also show that closed convex subsets of Banach spaces are weakly closed. We then introduce weak-* convergence and prove two powerful weak compactness theorems: Helly’s Theorem for weak-* convergence in the duals of separable Banach spaces and a weak sequential compactness theorem in reflexive Banach spaces.

This chapter outlines the purpose, scope, and structure of the book and introduces the scientific, data-driven approach to analysing and solving business problems and conducting business research.