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.

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.