Advanced Analysis II: Review of Double Integrals

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Explores the consequences of double integrals, including compact sets and continuity.

Explores the Fubini theorem on closed rectangles in R², discussing integrability, iterated integrals, and compact sets.

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Explores compact sets, extreme values, and function theorems on bounded sets.

Covers extreme values of continuous functions on compact sets and differentiability.

Covers the definitions and properties of double integrals over compact regions.

Covers normed spaces, dual spaces, Banach spaces, Hilbert spaces, weak and strong convergence, reflexive spaces, and the Hahn-Banach theorem.

Covers proper actions of groups on Riemann surfaces and introduces algebraic curves via square roots.

Covers the development of Taylor polynomials of order p around a point x.