MATH 454: Survey of Analysis
|Description: This course offers an introduction to several advanced topics in analysis: measure theory, functional analysis, and partial differential equations. It begins by discussing measurable sets and the construction of the Lebesgue integral in R^n. Following this several important function spaces are covered, including the L^p spaces. The notion of weak convergence will also be examined. These ideas are then applied to a few linear PDEs using the Lax–Milgram Theorem and basic ideas in the calculus of variations.
|Outcomes: Upon completion of this course students will have the ability to: understand the elements of Lebesgue measure and integration; state, prove, and apply the Monotone Convergence theorem, Fatou's Lemma, and Dominated Convergence theorem; understand and apply the fundamental theorem of calculus for Lebesgue integrals. They will also understand L^p spaces, including the Minkowski and Hölder inequalities. Finally, students will see how these concepts can be applied to study partial differential equations.