Provides the mathematical foundations for graduate-level economics, with applications illustrating each set of tools. Topics include real analysis and linear algebra (sequences, compactness, eigenvalues, quadratic forms, concavity, convexity); constrained and unconstrained optimization (Lagrange multipliers, Kuhn–Tucker conditions, constraint qualification) together with the implicit function and envelope theorems for comparative statics; separating and supporting hyperplane theorems and the theorem of the alternative (Farkas’ lemma); and correspondences, hemicontinuity, the theorem of the maximum, and fixed point theorems (Brouwer, Kakutani). These tools provide a foundation for studying graduate microeconomics, macroeconomics, and econometrics.