MATH 231 Lectures Log

Lecture 1, Aug 19

Topics

  • Introduction to the course.
  • Main object of the course: infinite-dimensional spaces.
  • How to see a PDE as an ODE on an infinite-dimensional space.

Readings

  • Read the whole Chapter 1 of the textbook.

Lecture 2, Aug 21

Topics

  • Why we need topology.
  • Topological vector spaces.
  • Topological vector spaces in finite dimension.
  • Definitions of:
    1. neighborhood of a point,
    2. continuity of a function,
    3. convergence of a sequence to a point,
    4. vector spaces,
    5. norm on a vector space,
    6. metric on a topological space.

Readings

  • K. Conrad (UConn), FINITE-DIMENSIONAL TOPOLOGICAL VECTOR SPACES.
    This document contains much more than we discussed, use it as a reference for the parts we covered and as a source of extra material in case you want to see a full proof of the uniqueness of Hausdorff vector spaces topologies in finite dimension.

Homework #1

Due date: Sep 6

  1. Let $\|\cdot\|$ be a norm on a vector space $V$. Prove that the function $d(u,v)=\|u-v\|$ is a metric on $V$.
  2. Prove that $\|(a,b)\|_1=|a|+|b|$ is a norm on $\Bbb R^2$ and draw the unit circle with respect to this norm.
  3. Let $\|(a,b)\|_2=\sqrt{|a|^2+|b|^2}$ be the Euclidean norm on $\Bbb R^2$. Prove directly that the topology induced on $\Bbb R^2$ by $\|\cdot\|_2$ coincides with the topology induced by $\|\cdot\|_1$.
  4. Draw in a single picture, using any mean you see fit, the "unit circles" centered at zero of the planar metrics induced by the following norms $\|(a,b)\|_p=(|a|^p+|b|^p)^{1/p}$ for $p=1,1.5,2,3,10$.

Lecture 3, Sep 2

Topics

  • Cauchy sequences in metric spaces.
  • Complete metric spaces.
  • Completing a non complete metric space.

Readings

Lecture 4, Sep 4

Topics

  • Continuing the proof of completion of a metric space.
  • Banach Spaces.
  • Several norms on the space of smooth functions.

Readings

Lecture 5, Sep 9

Topics

  • End of the proof of completion of a metric space.
  • Lebesgue integration obtained via completion of $C^\infty$ with respect to the $L^1$ norm.
  • $L^p$ spaces with $p\geq1$.
  • $L^p$ spaces with $p\in(0,1)$ (F-spaces).
  • Metric space structure of $C^\infty([0,1])$ (Fr´chet space).

Readings

Homework #2

Due date: Sep 14

Solve problems 1, 3, 4, 5 and 8 at this page.

Lecture 6, Sep 11

Topics

  • Hilbert Spaces

Readings

Lecture 7, Sep 16

Topics

  • Hilbert spaces
  • Mollifiers

Readings

Lecture 8, Sep 18

Topics

  • Differential functions

Readings

Lecture 9, Sep 23

Topics

  • Integrable functions.

Readings

Lecture 10, Sep 25 (video 1, video 2)

Topics

  • $L^p$ spaces with $p\in(0,1)$.
  • Integrable functions.

Readings

Homework #3

Due date: Sep 30

Solve problems 9-12 at this page.

Lecture 11, Sep 30

Topics

  • Bounded operators

Readings

Lecture 12, Oct 2

Topics

  • Uniform Boundedness principle
  • Open mapping theorem
  • Closed graph theorem

Readings

Lecture 13, Oct 7

Topics

  • Closed operators in Banach and Frechet spaces.

Readings

Lecture 14, Oct 9

Topics

  • Closable operators

Readings

Homework #4

Due date: Oct 19

Solve problems 6-11 at this page.

Lecture 15, Oct 14

Topics

  • The first order operator $2y\partial_x+(1-y^2)\partial_y$ on $L^1_{loc}({\mathbb R}^2)$.

Readings

Lecture 16, Oct 16

Topics

  • The Laplacian operator on $C^0({\mathbb R}^2)$ and on $L^2({\mathbb R}^2)$.

Readings

Lecture 17, Oct 21

Topics

  • The Laplacian operator on $L^1({\mathbb R}^2)$.
  • Compact operators

Lecture 18, Oct 23

Topics

  • Compact operators
  • Poisson's equation in one variable

Lecture 19, Oct 28

Topics

  • Poisson's equation in one variable
  • Solving some Functional Analysis problem

Lecture 20, Oct 30

Topics

  • Solving some Functional Analysis problem
  • The Hahn-Banach theorem

Lecture 21, Nov 4

Topics

  • Zorn's Lemma
  • Ordinals and transfinite induction
  • Proof of the Hahn-Banach theorem
  • Some important corollaries of the Hahn-Banach theorem

Readings

Lecture 22, Nov 6

Topics

  • Topology of dual spaces
  • Some examples of dual spaces
  • Reflexive spaces
  • Weak topologies

Readings

Lecture 23, Nov 13

Topics

  • Extension of a bounded linear functional defined on a proper subspace
  • Dual of a Hilbert space

Readings

Lecture 24, Nov 18

Topics

  • Adjoint of an operator.
  • Dual of $L^p_loc(\mathbb R)$.
  • Dual of $C^\infty_c(\mathbb R)$.

Readings

Lecture 25, Nov 20

Topics

  • Distributional derivative of the step function.
  • Distributional derivative of the Dirac delta.
  • Order of a distribution.
  • Dual of $C^\infty(\mathbb R)$.

Readings

Lecture 26, Nov 25

Topics

  • Weak convergence.
  • Weak$^*$ convergence

Readings

Lecture 27, Nov 26

Topics

  • Weak convergence.
  • Weak$^*$ convergence

Readings