This chapter reviews essential concepts from real and functional analysis for studying fluid-structure interaction (FSI). It begins with Banach and Hilbert spaces, emphasizing \({\mathbf {L}}^{\mathrm {p}}\) ? and Sobolev spaces, and explores the Fourier transform in \({\mathbf {L}}^{2}(\mathbb {R}^{\mathrm {d}}\) ). Next, it introduces Bochner spaces (e.g., \({\mathbf {L}}^{\mathrm {p}}(\mathbf {0,T};\ \mathbf {B})\) ) for time-dependent PDEs and extends these to moving domains, addressing time-dependent geometries and key inequalities. The chapter then covers compact embeddings and foundational equations of fluid dynamics and elasticity, including a priori estimates and weak solutions. Finally, it introduces constructive methods—Rothe’s method and Lie operator splitting—for approximating solutions to PDEs and FSI problems. These tools are crucial for analyzing evolutionary PDEs and dynamic domain problems.