Real Analysis
Complete course reference covering topological and metric spaces, continuity, compactness, completeness, and series of functions. Worked problem sets, past tests, and final exams.
11 chapters building from foundational topology and metric spaces through convergence, compactness, and function spaces.
The underlying spaces — topology and metric — that everything else lives in
The foundational structure of analysis: open sets, closed sets, and the topology axioms. Includes the discrete, trivial, and Euclidean topologies.
Spaces equipped with a notion of distance. The metric gives rise to a topology through open balls. Includes standard examples, norms, and the metric topology.
Interior, closure, continuity, compactness, and connectedness
Operations that capture the "inside," "including the edge," and the "edge itself" of a set. The interior is the largest open subset; the closure is the smallest closed superset.
The topological definition of continuity: preimages of open sets are open. Equivalent characterizations via sequences and epsilon-delta on metric spaces.
The generalization of "finite" for infinite sets. Every open cover has a finite subcover. In metric spaces: equivalent to sequential compactness and total boundedness. The Heine-Borel theorem.
Topological notion of being "in one piece." Connected sets cannot be split into two disjoint nonempty open subsets. Path-connectedness is a stronger, more intuitive form.
Sequences, Cauchy sequences, and completeness
Convergence of sequences in topological and metric spaces. Subsequences, accumulation points, and the connection between sequences and closed sets.
Cauchy sequences capture the intuitive notion of "eventually close" without requiring a limit. Complete spaces are those in which every Cauchy sequence converges.
Taylor expansion in several variables and constrained optimization
Partial derivatives, the gradient, and the total derivative for functions of several variables. Clairaut's theorem on equality of mixed partials. Classes of differentiable functions (C^k).
When can an implicit equation F(x, y) = 0 be solved locally for y as a function of x? The implicit function theorem gives sufficient conditions via the non-singular Jacobian. Inverse function theorem as a special case.
Taylor expansions for functions of several variables, with integral and Lagrange remainders. Critical points, the Hessian matrix, and the second derivative test for classifying extrema.
Optimizing a function subject to equality constraints. The method of Lagrange multipliers, derived via the implicit function theorem. Extends to multiple constraints with a full-rank Jacobian condition.
Riemann and Lebesgue integration, measure theory
Past tests and final examinations