Integral Properties on Closed Pavés

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

Covers the concept of compact embedding in Banach spaces and Sobolev inequalities.

Explores continuous functions on compact sets, discussing boundedness and extremum values.

Covers the preliminaries in measure theory, including loc comp, separable, complete metric space, and tightness concepts.

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

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

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

Explores harmonic forms on Riemann surfaces, covering uniqueness of solutions and the Riemann bilinear identity.

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

Covers the topology of Riemann surfaces and the concept of triangulation using finitely many triangles.