TOPICS

PART I: Topology

Normed vector spaces

Continuity of maps

Compactness

Connectedness

Spaces of continuous functions

Supplementary handouts (for advanced students):

(adapted from more advanced classes and not yet in final form)

Definitions and Theorems from General Topology

References: [D]= Topology, by James Dugundji; [R]=Functional Analysis, by W. Rudin

Locally compact Banach spaces are finite dimensional (includes 4 problems)

Spaces of Continuous Functions (outdated)

Stone-Weierstrass theorem-notes (includes 6 problems)

Ascoli-Arzela-Notes (final-included 7 exercises with solutions, and 11 extra problems.)

PART 2: Differentiable functions

Directional and partial derivatives

Differentiable functions

Functions of class C^k, Taylor's theorem

Relative extrema

Convex functions

Spaces of differentiable functions

PART 3: Differentiable maps and surfaces

Differentiable maps

Compositions and the Chain Rule

Inverse Function Theorem

Implicit Function Theorem

Surfaces in R^n

Constrained extrema (Lagrange multipliers)