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.

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.

Problems, that is ‘questions’, drive research. Without research questions there would hardly be any research at all. Research problems are not ‘given’, however; they are detected and constructed. How research problems are captured and framed drives subsequent research activities. In normal research situations, we first select a topic and then formulate a research problem within that topic. The process of constructing a research problem is not straightforward and often involves a lot of back-and-forth adjustments and refinement. In this chapter we particularly focus on how to construct and adequately capture research problems. The role of reviewing past literature to identify weaknesses and gaps is also examined.

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.

By this point we have established the need for mixers in a radio, which is to provide frequency translation, and consequently to ease analog and digital signal processing by means of performing them at a conveniently lower frequency. Since the frequency translation is created due to either time variance or nonlinearity, or often both, the small signal analysis performed typically on linear amplifiers does not hold. This makes understanding and analysis of the mixers somewhat more difficult. Although exact methods have been presented, in this chapter we resort to more intuitive yet less complex means of analyzing the mixers. In most cases this leads to sufficient accuracy but more physical understanding of the circuit.

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.

The most commonly used technique for the analysis of quantitative data in business research is multiple regression analysis. This is a powerful technique for understanding the relationships between variables, which variables have the most impact, and for prediction. In this chapter, we consider how to specify regression models, how to estimate the models, and how to use the estimated models to undertake some simple hypothesis tests. We emphasize that the researcher has to exercise his/her judgement in deciding not only the specification of the initial model but also in how to adapt and interpret the model in response to the various statistical tests.

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.

As a corollary of the Hahn-Banach Theorem, we show that any two convex sets can be separated using a linear functional; a key ingredient is the definition of the Minkowski functional of a convex set. This separation theorem allows us to give a characterisation of convex sets in terms of their supporting hyperplanes that will be useful later. We then define the closed convex hull of a set, introduce the notion of extreme points in a convex set, and prove the Krein-Milman Theorem: a non-empty compact convex subset of a Banach space is the closed convex hull of its extreme points.

In this chapter we review some of the basic concepts in communication systems. We start with a brief summary of Fourier and Hilbert transforms, both of which serve as great tools for analyzing RF circuits and systems. We also present an overview of network functions and the significance of poles and zeros in circuits and systems. To establish a foundation for the noise analysis presented in Chapter 5, we also provide a brief summary of stochastic processes and random variables. We conclude this chapter by briefly describing the fundamentals of analog modulation schemes and analog modulators.

Problems, that is ‘questions’, drive research. Without research questions there would hardly be any research at all. Research problems are not ‘given’, however; they are detected and constructed. How research problems are captured and framed drives subsequent research activities. In normal research situations, we first select a topic and then formulate a research problem within that topic. The process of constructing a research problem is not straightforward and often involves a lot of back-and-forth adjustments and refinement. In this chapter we particularly focus on how to construct and adequately capture research problems. The role of reviewing past literature to identify weaknesses and gaps is also examined.

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.

Once again we use the Baire Category Theorem to prove results about linear maps between Banach spaces. We prove the Open Mapping Theorem and, as a corollary, the Inverse Mapping Theorem, which allows for some simplification in the spectral theory of bounded operators. As an application, we prove the existence of a ‘basis constant’ for any Schauder basis in a separable Banach space. Finally, we use the Inverse Mapping Theorem to prove the Closed Graph Theorem, which gives an alternative way to check whether a linear map T from X into Y is bounded, provided both X and Y are Banach spaces.

We return to the topic of dual spaces. We prove Young’s inequality and Hölder’s inequality in the l^p and L^p spaces. We identify the dual spaces of l^p and L^p up to isometric isomorphisms: the proofs for l^p are presented in full, with the measure-theoretic proofs for L^p contained in Appendix B.