Using the Riesz Representation Theorem, we define the Hilbert adjoint T^* of a linear map T from H into K (when H and K are both Hilbert spaces). This is another linear map from K into H. We show that the norm of T and its adjoint are equal. An operator is self-adjoint if it is equal to its adjoint (T=T^*); we compute explicitly the adjoints of some simple linear operators and give conditions under which they are self-adjoint.

We prove results about the spectrum of compact operators on Banach spaces, recovering many of the results obtained earlier for compact self-adjoint operators on Hilbert spaces. We show that the spectrum consists entirely of eigenvalues, apart perhaps from zero, that each eigenvalues has finite multiplicity, and that the eigenvalues have no accumulation points except zero.

Empirical research requires the collection and analysis of data and other information. The quality of the research (and the conclusions derived therefrom) depend upon the collection of appropriate data, the quality of the data collected, and on how well the data are analysed. Quantitative research requires the measurement and enumeration of the variables to be used in the analysis. In this chapter, we first explain the process of operationalization, by which researchers decide how to measure the theoretical concepts they use. The second section considers different scales of measurement, and highlights some of the implications for empirical analysis. The third section focuses on the measurement of multi-dimensional variables, and the generation of latent constructs. The fourth section addresses how to assess the reliability and validity of variables and multi-dimensional constructs. The fifth section offers some practical suggestions for improving the measurement of the variables used in quantitative research, whilst the final section is concerned with measurements in qualitative research.

In cases of international or cross-cultural research we need to take extra care at every stage of the process, and this chapter looks at various aspects of this. Research involving unfamiliar environmental and cultural differences may complicate the understanding of the research problem, and researchers often fail to anticipate the impact of local cultures on the question asked. Consideration also needs to be given to the scope and limits of the problem. In some cultures, a broader scope is necessary to cover the necessary variables. Comparability of data is the main issue in international/cross-cultural research, and it is not possible to use data gathered in one market for another market. This is due not just to the availability and reliability of data but also to the manner in which data are collected and analysed.

We introduce inner product spaces. After proving the Cauchy-Schwarz inequality we show that any inner product induces a norm and that the norm then satisfies the parallelogram identity. We show that the inner product can be recovered from the induced norm (via the polarisation identity). We define Hilbert spaces as complete inner product spaces and show that the spaces l^2 and L^2 are Hilbert spaces.

By now we have a general idea of what the receivers and transmitter look like. Nonetheless, the exact arrangement of the blocks, the frequency planning involved, the capabilities of digital signal processing, and other related concerns result in several different choices that are mostly application dependent. While our general goal is to ultimately meet the standard requirements, arriving at the proper architecture is largely determined by the cost and power consumption concerns. The goal of this chapter is to highlight these trade-offs, and present the right architecture for a given application, considering the noise, linearity, and cost trade-offs, which were generally described in Chapters 5 and 6. Moreover, we will see that the proper arrangement of the building blocks is a direct function of the circuit capabilities that we presented in Chapters 7–11.

The Hahn-Banach Theorem allows for the extension of linear maps defined on subspaces of normed spaces to the whole space in a way that respects sublinear bounds. The simplest case is the extension of bounded linear functionals from subspaces to the whole space. We prove this result here, first for real spaces and then for complex spaces. The proof requires use of Zorn’s Lemma, unless we assume that the underlying space is separable.

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 research design is the overall plan for relating the conceptual research problem to relevant and practicable empirical research. In other words, the research design provides a plan or a framework for data collection and its analysis. It reveals the type of research (e.g. exploratory, descriptive, or causal) and the priorities of the researcher. The research methods, on the other hand, refer to the techniques used to collect and analyse data. This chapter looks at a variety of research designs and methods, as well as at the concept of validity.

In Chapter 5 we showed that according to the Friis equation the lower limit to a receiver noise figure is set by the first block. Moreover, the gain of the first stage helps reduce the noise contribution of the subsequent stages. Hence, it is natural to consider a low-noise amplifier at the very input of a receiver. A first look then will tell us that for a given application, the higher the gain of the amplifier the better. However, the limit to that is typically imposed by the distortion caused by the blocks following the low-noise amplifier. As we stated earlier, there is a compromise between the required gain, noise, and linearity for a given application, and a certain cost budget. These requirements may be different for a different application or standard. In this chapter we assume that these requirements are given to us, and our goal is to understand the design trade-offs. In Chapter 12 we will study receiver architectures and various trade-offs associated with each choice of architecture.

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.

The research design is the overall plan for relating the conceptual research problem to relevant and practicable empirical research. In other words, the research design provides a plan or a framework for data collection and its analysis. It reveals the type of research (e.g. exploratory, descriptive, or causal) and the priorities of the researcher. The research methods, on the other hand, refer to the techniques used to collect and analyse data. This chapter looks at a variety of research designs and methods, as well as at the concept of validity.

We give some important applications of the Hahn-Banach Theorem. We prove the existence of a support functional and hence that X^* separates points in X. Then we prove the existence of a functional that encodes the distance from a linear subspace, which is an important ingredient in a number of subsequent proofs. We show that separability of X^* implies separability of X, define the Banach adjoint of a linear map (between Banach spaces), and prove the existence of ‘generalised Banach limits’.

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.

We define the notion of a norm and a normed space. We prove that various canonical definitions are indeed norms (e.g. the l^p norm, the L^p norm, and the supremum norm). We discuss convergence, equivalent norms, and various notions of isomorphism between normed spaces. Finally, we discuss separability in more detail.