Week 1: Normed linear spaces, examples. Continuous linear transformations, examples.
Week 2: Continuous linear transformations. Hahn-Banach theorem-extension form. Reflexivity.
Week 3: Hahn-Banach theorem-geometric form. Vector valued integration.
Week 4: Baire’s theorem,.Principle of uniform boundedness. Application to Fourier series. Open mapping and closed graph theorems.
Week 5: Annihilators. Complemented subspaces. Unbounded operators, Adjoints.
Week 6: Weak topology. Weak-* topology. Banach-Alaoglu theorem. Reflexive spaces.
Week 7: Separable spaces, Uniformly convex spaces, applications to calculus of variations.
Week 8: L^p spaces. Duality, Riesz representation theorem.
Week 9: L^p spaces on Euclidean domains,.Convolutions. Riesz representation theorem.
Week 10: Hilbert spaces. Duality, Riesz representation theorem. Application to the calculus of variations. Lax-Milgram lemma. Orthonormal sets.
Week 11: Bessel’s inequality, orthonormal bases, Parseval identity, abstract Fourier series. Spectrum of an operator.
Week 12: Compact operators, Riesz-Fredholm theory. Spectrum of a compact operator. Spectrum of a compact self-adjont operator.