Topics in Analysis

Reference: Modern Real Analysis, Second Edition, by W. Ziemer (with Monica Torres).

Lecture 1-4: Lp spaces and the space C(X) are Banach spaces. Bounded linear mappings. Dual spaces. Hahn-Banach Theorem. The Open Mapping Theorem.The Closed Graph Theorem. Separable spaces. Reflexive spaces. L^p, 1. Duality in L^p. L^1 is not reflexive.

Lecture 5: Weak topology. The unit ball in a reflexive Banach space X is sequentially compact in the weak topology.

Lecture 6: The heat equation and the Laplace's equation(derivation).

Lecture 7: Weak* topology. Alaouglu's Theorem. Weak convergence in Lp.

Lecture 8: Weak Convergence. Heat equation.

Lecture 9: Hilbert space. Riesz Representation Theorem for Hilbert Spaces.

Lecture 10: Riesz Representation Theorem in L^p(X). Radon-Nikodym Theorem.

Lecture 11: Riesz Representation Theorem (local version) (I).

Lecture 12: Riesz Representation Theorem (local version) (II).

Lecture 13 Riesz Representation Theorem (local version) (III).

Lecture 14: Riesz Representation Theorem (local version) (IV).

Lecture 15: Some examples of PDE. Weak Convergence in L^p and Partial Differential Equations.

Lecture 16: Hiperbolic Systems of Conversation Laws: An Application of the Riesz Representation Theorem.

Lecture 17: An Introduction to Distributions.

Lecture 18: Properties of Convolutions (I).

Lecture 19: Properties of Convolutions (II). Definition of Distribution.

Lecture 20: Notes on the Total Variation of a measure. Proof of the Riesz Representation Theorem (global version). Positive Distributions.

Lecture 21: Diferentiation of Distributions.

Lecture 22: Functions of bounded variation (in 1 dimension) and distributions.

Lecture 23: Functions with finite essential variation. The total variation of a measure. The Riesz Representation Theorem (compact version)

Lecture 24: Functions of bounded variation in many dimensions. Differentiability of functions of several variables.

Lecture 25: Rademachers's Theorem for Lipschitz functions.

Lecture 26: Proof of Rademacher's Theorem (continuation).

Lecture 27: Change of Variable Formula.

Lecture 28: Proof of the Area Formula

Lecture 29: Sobolev spaces. Aproximation of Sobolev Functions (I).

Lecture 30: Representative functions in Wo^{1,p}.

Lecture 31: Aproximation of Sobolev Functions (II).

Lecture 32: Traces of Sobolev Functions. The space Wo^{1,p}.

Lecture 33: Proof of the Sobolev. Imbbeding Theorem.

Lecture 34: Laplace's equation.

Lecture 35: Proof of existence of weakly harmonic functions.

Lecture 36: Weakly harmonic functions are continuous.

Lecture 37: Notes on the area and coarea formulas. The minimal surface equation.

Lecture 38: Continuity of weakly harmonic functions (Part II)

Lecture 39: Weakly harmonic functions are infinitely differentiable.