This course will cover some of the basic tools of analysis that are extremely useful in many areas of mathematics, including PDE's, stochastic analysis, harmonic analysis and complex analysis. Specific topics covered in the course include: Geometric lemmas (Vitali, Wiener, etc.) and geometric decomposition theorems (Whitney, etc.) and their applications to differentiation theory and to the Hardy-Littlewood maximal function; convolutions; approximations to the identity and their applications to boundary value problems in R^d with L^p-data; the Fourier transform and its basic properties on L¹ and L² (including Plancherel's theorem); interpolation theorems for linear operators (Marcinkiewicz, Riesz-Thorin); the basic Calderón-Zygmund singular integral theory and some of its applications; the Hardy-Littlewood-Sobolev inequalities for fractional integration and powers of the Laplacian and other elliptic operators; the inequalities of Nash and Sobolev viewed from the point of the heat semigroup.