This textbook is the first of a two-part set providing a thorough-yet-accessible introduction to the subject of function spaces. In this first volume, spaces of continuous and integrable functions are covered in detail.

Starting from the familiar notions of continuous and smooth functions, the volume gradually progresses to advanced aspects of Lebesgue spaces and their relatives. Key aspects such as basic functional analytic properties, weak convergence and compactness are covered in detail, concluding with an introduction to the fundamentals of real and harmonic analysis. Throughout, the authors provide helpful motivation for the underlying concepts, which they illustrate with selected applications, demonstrating the relevance and practical use of function spaces.

Designed with the student in mind, this self-contained volume offers multiple syllabi, guiding readers from elementary properties to advanced topics. Visual learners will appreciate the inclusion of figures summarising key outcomes, proof strategies, and 'metaprinciples'. Assuming only multivariable calculus and elementary functional analysis, as conveniently summarised in the first chapters, this volume is designed for lecture courses at the graduate level and will also be a valuable companion for young researchers in analysis.

Dominic Breit is currently Associate Professor (Reader) in the Department of Mathematics at Heriot-Watt University, Edinburgh. His research interests range from pure topics such as Sobolev spaces, regularity theory for nonlinear PDEs and the calculus of variations, to applications in fluid mechanics, in particular compressible fluids, stochastic Navier-Stokes equations and fluid-structure interaction.

Franz Gmeineder holds the tenure-track professorship "Theory of Partial Differential Equations" at the Department of Mathematics at the University of Konstanz, Germany. His research interests are centered around regularity theory in the Calculus of Variations, with a particular focus on linear growth functionals, real analysis and L1-based function spaces.

This textbook provides a thorough-yet-accessible introduction to function spaces, through the central concepts of integrability, weakly differentiability and fractionally differentiability.

In an essentially self-contained treatment the reader is introduced to Lebesgue, Sobolev and BV-spaces, before being guided through various generalisations such as Bessel-potential spaces, fractional Sobolev spaces and Besov spaces. Written with the student in mind, the book gradually proceeds from elementary properties to more advanced topics such as lower dimensional trace embeddings, fine properties and approximate differentiability, incorporating recent approaches. Throughout, the authors provide careful motivation for the underlying concepts, which they illustrate with selected applications from partial differential equations, demonstrating the relevance and practical use of function spaces.

Assuming only multivariable calculus and elementary functional analysis, as conveniently summarised in the opening chapters, A Course in Function Spaces is designed for lecture courses at the graduate level and will also be a valuable companion for young researchers in analysis.

1 Introduction.- 2 Preliminaries I: Calculus and Measure Theory.- 3 Preliminaries II: Functional Analysis.- 4 Spaces of Continuous Functions.- 5 L^p-Spaces.- 6 Basics From Real and Harmonic Analysis.- 7 Weakly Differentiable Functions.- 8 Embeddings on Rⁿ.- 9 Traces, Extensions and Embeddings on Domains.- 10 Potential Spaces and Fractional Sobolev Spaces.- 11 Fine Properties of Sobolev Functions.- 12 Fine Properties of Functions of Bounded Variation.- 13 Fractional Differentiability: Besov Spaces.- 14 Outlook.- Bibliography.- Index.