We discuss how to define a basis for a general normed space (a ‘Schauder basis‘). We then consider orthonormal sets in inner-product spaces and orthonormal bases for separable Hilbert spaces. We give a number of conditions that ensure that a particular orthonormal sequence forms an orthonormal basis, and as an example, we discuss the L^2 convergence of Fourier series.

We consider linear maps between normed spaces. We define what it means for a linear map to be bounded and show that this is equivalent to continuity. We define the norm of a linear operator and show that the space of all linear maps from X to Y is a vector space, which is complete if Y is complete. We give a number of examples and then discuss inverses and invertibility in some detail.

In this chapter, we first provide a detailed discussion of the advantages and disadvantages of collecting and using secondary data, and highlight some important secondary data sources. The next section then considers the advantages and disadvantages of collecting and using primary data. The following three sections are devoted to sampling. With secondary data, the researcher is obliged to accept the data that are publicly available, and is not able to influence how the data are collected or how much data are collected. In contrast, the researcher collecting primary data needs to decide whether to survey the entire population or just a sample, to choose an appropriate sampling procedure, and to determine the sample size that will assure a satisfactory level of precision in the subsequent empirical analysis. The final two sections are then devoted to undertaking the two most common methods of primary data collection, namely questionnaire surveys and experiments.

A huge array of statistical methods are available to the researcher, of variable levels of sophistication, and a comprehensive survey would be well beyond the scope of this textbook. Here we outline three methods which are widely used in business studies research, namely factor analysis, structural equation modelling, and event study analysis. In each case, we explain the key elements of each method, the underlying intuition, and how to interpret the results, and then provide an example from the business literature.

We investigate finite-dimensional normed spaces. We show that in a finite-dimensional space, all norms are equivalent, and that being compact is the same as being closed and bounded. We also show that a normed space is finite-dimensional if and only if its closed unit ball is compact, using Reisz’s Lemma.

Almost every radio includes one or more filters for the general purpose of separating a useful information-bearing signal from unwanted signals such as noise or interferers. We will discuss the role of filters and their requirements in Chapter 6 when we talk about distortion. Here we discuss their properties from a circuit point of view and offer general design guidelines.

We consider the existence of closest points in convex subsets of Hilbert sapces. In particular this enables us to define the orthogonal projection onto a closed linear subspace U of a Hilbert space H, and thereby decompose any element x of H as x=u+v, where u is an element of U and v is contained in its orthogonal complement. We also discuss ‘best approximations’ of elements of H in spaces spanned by collections of elements of H.

We show that the space C^0 is not complete in the L^1 norm and use this to motivate the abstract completion of a normed space and the Lebesgue integral. We use this approach to define the family L^p of Lebesgue spaces as the completion of the space of continuous functions in the L^p norm, and we prove some properties of these spaces.

In this chapter basic components used in RF design are discussed. Detailed modeling and analysis of MOS transistors at high frequency can be found already in many analog books [1], [2]. Although mainly offered for analog and high-speed circuits, the model is good enough for most RF applications operating at several GHz and beyond, especially for nanometer CMOS processes used today. Thus, instead we will have a more detailed look at inductors, capacitors, and LC resonators in this chapter. We will also briefly discuss the fundamental operation of distributed circuits and transmission lines and follow up with more in Chapter 3. In Chapters 5 and 7 we will discuss some of the RF aspects of the transistors, including a more detailed noise analysis as well as substrate and gate resistance. New to this edition are Sections 1.8 and 1.9, which cover the fundamentals of wave propagation and antennas.

After completing the data collection and analysis, the research problem, the data collected, and the findings need to be presented in a logical, consistent, and persuasive report. This chapters outlines a typical format for such a research report, and describes the contents of each section. It also discusses oral presentations and writing for publication.

This chapter explains what we mean by research in business studies and to discuss differences between systematic research and common sense or practical problem solving. It looks at what we mean by knowledge and why we do research, examining different research orientations and approaches and the influence of the researcher’s background and basic beliefs concerning research methods and processes. We stress the importance of learning to think and work systematically and developing analytical capabilities in order to produce accurate and reliable results. We also discuss researchers’ moral responsibility towards both their subjects and the readers of their reports.

In this chapter we present the phase-locked loops and synthesizers, built upon the discussions of the previous chapter, VCOs and crystal oscillators. It is a new chapter in this edition, although some pieces of it existed in the previous edition under Chapter 8. A detailed discussion of PLLs and synthesizers would perhaps require an entire book of its own, and our objective here is only to establish enough background to allow design and analysis of synthesizers for typical radio applications.

There is a canonical way to associate an element of a Banach space X with its second dual (‘double dual’) space X^{**}. If this mapping is onto, then X is said to be reflexive. We show that Hilbert spaces, l^p and L^p for 1<p<infty, are reflexive. We then prove some general properties of reflexivity, in particular that X is reflexive if and only if X^* is reflexive, and that reflexivity is inherited by subspaces.