In this chapter, we explore the foundational role of linear operators and functionals in the development of modern mathematical frameworks such as distributions or generalized functions. Starting with an introduction to bounded linear functionals and the Riesz representation theorem, we establish the core principles that lead to the formalization of distributions. Using the intuitive concept of integration by parts, we extend the definition of the derivative to include discontinuous functions, with particular attention to the Dirac delta function. We then formally define distributions as linear functionals acting on a space of test functions and demonstrate how they can be used to represent images in image processing. The final section of the chapter focuses on Fourier analysis, applying the theory to problems such as convolution and Fourier transforms, offering practical tools for solving real-world imaging problems.