Compact Embedding: Theorem and Sobolev Inequalities

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

Introduces linear and bounded operators, compact operators, and the Banach space.

Explores Banach spaces, emphasizing reflexivity and sequence convergence in a rigorous mathematical framework.

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

Explores interior points, boundaries, adherence, and compact sets, including definitions and examples.

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

Explores compact subsets of R^n, convergence theorems, and set properties.

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

Explores the proof of Weyl's theorem, focusing on discrete spectrum, ground states, and potential energy continuity.

Covers weak derivatives, their properties, and applications in functional analysis.