It is one of the major results of finite-dimensional linear algebra that all the eigenvalues of real symmetric matrices are real and that the eigenvectors of distinct eigenvalues are orthogonal. In this chapter we prove similar results for compact self-adjoint operators on infinite-dimensional HIlbert space: we show that the spectrum consists entirely of real eigenvalues (except perhaps zero), that the multiplicity of every non-zero eigenvalue is finite, and that the eigenvectors form an orthonormal basis for H. The last fact follows from the Hilbert-Schmidt Theorem, which allows us to write such operators in terms of their eigenvalues and eigenvectors.

In business studies most researchers need to collect some primary data to answer their research question. This entails deciding what kind of data collection method to use, which depends upon an overall judgement on which type of data is needed for a particular research problem. One important aspect is to identify the scope of the study and unit of analysis and what type of analysis is needed. After looking briefly at the chief differences between quantitative and qualitative approaches, the chapter looks at different qualitative methods and when to use them.

Qualitative research imposes specific analytical challenges. This chapter addresses important characteristics of qualitative research and qualitative data. Strategies and procedures to handle the analytical challenges are also dealt with, as well as validity and reliability issues in qualitative research.

This chapter deals with some conceptual (theoretical) foundations of research. Practical business research is often thought of as collecting data from various statistical publications, constructing questionnaires, and analysing data by using computers. Research, however, also comprises a variety of important, non-empirical tasks, such as finding/‘constructing’ a precise problem, and developing perspectives or models to represent the problem under scrutiny. In fact, such aspects of research are often the most crucial and skill demanding. The quality of the work done at the conceptual (theoretical) level largely determines the quality of the final empirical research. This is also the case in practical business research. Important topics focused on in this chapter are the research process and the role of concepts and theory.

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.

In this chapter we study the challenging problem of delivering power to antenna efficiently. The linear amplifier topologies that we have discussed thus far are fundamentally incapable of achieving high efficiency. Considering the high demand for improving battery life, this shortcoming becomes very critical when delivering hundreds of mW or several watts of power into the antenna. This issue is exacerbated in most modern radios that employ complex modulation schemes to improve the throughput without raising the bandwidth. As we discussed in Chapter 6, such systems demand more linearity on the power amplifier, and hence achieving a respectable efficiency becomes more of a challenge.

We prove some key results about spaces of continuous functions. First we show that continuous functions on an interval can be uniformly approximated by polynomials (the Weierstrass Approximation Theorem), which has interesting applications to Fourier series. Then we prove the Stone-Weierestrass Theorem, which generalises this to continuous functions on compact metric spaces and other collections of approximating functions. We end with a proof of the Arzelà-Ascoli Theorem.

In business studies most researchers need to collect some primary data to answer their research question. This entails deciding what kind of data collection method to use, which depends upon an overall judgement on which type of data is needed for a particular research problem. One important aspect is to identify the scope of the study and unit of analysis and what type of analysis is needed. After looking briefly at the chief differences between quantitative and qualitative approaches, the chapter looks at different qualitative methods and when to use them.

Qualitative research imposes specific analytical challenges. This chapter addresses important characteristics of qualitative research and qualitative data. Strategies and procedures to handle the analytical challenges are also dealt with, as well as validity and reliability issues in qualitative research.

We introduce the property of completeness and prove some abstract results about complete normed spaces (Banach spaces). We then give a number of examples of Banach spaces: l^p, L^p, and spaces of continuous functions. We then discuss convergence of series in Banach spaces and end the chapter with a proof of the Contraction Mapping Theorem.

Using the Baire Category Theorem, we prove the Principle of Uniform Boundedness, which allows us to deduce uniform bounds on collections of bounded linear operators from pointwise properties. We use the powerful corollary known as the “Condensation of Singularities” to show that there are continuous periodic functions whose Fourier series do not converge pointwise everywhere.