- This course is the continuation of APPM 5440, with the same instructor and same book. The biggest change is that we are no longer using D2L for grades and homework solutions; instead, we are using canvas. Visit our course's canvas website.
- We cover chapters 7, 8, 9, 11, and 12 of Hunter and Nachtergaele's book (APPM 5440 covered chapters 1--6), skipping chapters 10 and 13.

- The class meets in ECCR 131 every Monday/Wednesday/Friday from 1 PM to 1:50 PM
- Note: this is
*not*the same time/location as last semester

- Note: this is
- The final exam is Sunday May 6 from 7:30 PM to 10 PM
- If you have any issues with this date, let the instructor know immediately
- Since you have months of advance warning, there are very few valid reasons to miss it!

- Dropping the course
- After Jan 31 2018, if you drop the course, a "W" shows up on the transcript and you are charged for the class (last day to
*add*a class is Jan 24 2018) - After Mar 23 2018, the course cannot be dropped automatically online; you need to petition the dean for approval (which is unlikely to happen except in extenuating circumstances).

- After Jan 31 2018, if you drop the course, a "W" shows up on the transcript and you are charged for the class (last day to
- Exams from previous years:
- for both 5440 and 5450, they are collected at this website

**Quick Links**

**Text:** *Applied Analysis* by by J. Hunter and B. Nachtergaele, World Scientific Publishing, 1st ed., 2001, ISBN 978-9812705433.

**Syllabus:** APPM 5450 syllabus (last updated 1/21/18)

**Prelim exam**: this course is designed to partially prepare you for the Analysis prelim exam (Analysis prelim exam, old website) for graduate students in the applied math department (you should also know advanced calculus and the first semester of this course; in addition, independent studying is recommended).

Instructor | Room Number | Time |
---|---|---|

Stephen Becker | ECCR 131 | MWF 1 to 1:50 |

Instructor/TA | Room Number | Office Hours |
---|---|---|

Stephen Becker | ECOT 231 | Tues 3-3:30 PM; Wed 4:30-5 PM; Thurs 1-2 and 4-5 PM |

**Homework solutions are to uploaded to Canvas (canvas.colorado.edu) **

Homework | Due date |
---|---|

HW 1, ch 7 | Friday, Jan 26 2018 |

HW 2, ch 7,8 | Friday, Feb 2 2018 |

HW 3, ch 8 | Friday, Feb 9 2018 |

HW 4, ch 8 | Monday, Feb 19 2018 |

HW 5, ch 9 | Friday, Mar 2 2018 |

HW 6, ch 9 | Friday, Mar 9 2018 |

HW 7, ch 9 | Monday, Mar 19 2018 |

HW 8, ch 11 | Friday, Mar 23 2018 |

HW 9, ch 11 | Monday, April 9 2018 |

HW 10, ch 12 | Friday, April 13 2018 |

HW 11, ch 12 | Monday, April 23 2018 |

HW 12, ch 12 | Friday, April 27 2018 |

HW 13, ch 12 | HW is optional |

There are no mid-term exams, only a final (in-class). Half of the homework are "no collaboration" homeworks, which serve as mid-term exam scores.

The final exam is in accordance with the standard CU Final Exam schedule and is Sunday, May 6 2018 from 7:30 PM to 10:00 PM

**Handouts for 5450:**

- Crowd sourced solutions -- these are overleaf documents that the entire class can edit (solutions NOT by professor)
- Note: these are the read-only overleaf links, email professor to get edit links
- HW 9 extra problems
- Solutions to Jan 2015 Math dept prelim

- Absolutely continuous measures, and Radon-Nikodym Theorem (April 21 2018; 2018 version)
- includes MATH prelim practice
- Absolutely continuous measures, and Radon-Nikodym Theorem (2016 version)

- Hahn-Banach vs BLT theorems
- Convergence of Fourier series (updated version of 2016 version)
- A few facts about projections and orthogonal projections (Fri, Feb 2 2018)
- A few facts about set theory (Wed, Feb 7 2018)
- Proof of the facts from Terence Tao's notes (about uniform/strong/weak operator convergence) that we used in HW 4 (handwritten; Fri Feb 16 2018)
- Prop. 8.44 from the book, and extensions (Kadeck property, uniformly convex Banach spaces); (handwritten, Mon Feb 19 2018)
- See links below (from 2016 version) for pdf versions of the "pencasts"

some handouts from the 2016 version of 5450:

- For Monday Jan 25 2016, please watch the following pencast lecture on convolution (about 10 min)
- 2018 update: the LiveScribe web player isn't working, so try this static pdf version of the lecture on convolution

- For Wednesday Feb 24 2016, please watch the following:
- pencast on proof of Prop. 9.6 (10 min; spectrum lies within a ball of radius given by the norm of the operator)
- pencast on proof of Prop. 9.7, part 1 (11 min) and pencast on proof of Prop. 9.7, part 2 (10 min) (spectral radius)
- 2018 update: the LiveScribe web player isn't working, so try this static pdf version of the notes on prop 9.6 and 9.7

- Final exam review information
- List of topics for final, and short review of entire semester and long review of entire semester (put together by last year's TA)
- The APPM 5450 exam and APPM 5450 exam solutions (both from spring 2014 class). PLEASE do not look at solutions until you attempt the problems!
- The APPM 5450 exam and APPM 5450 exam solutions (both from spring 2015 class)
- The APPM 5440 exam and APPM 5440 exam solutions (both from fall 2014 class)
- The exam does not officially cover material from 5440, but the material naturally builds on the first semester, so you should know 5440 material
- The exam is 2.5 hours, closed book (similar to the prelim)

- Monday April 25 2016, here are some handwritten notes on ch 12 so you can take fewer notes in class
- Pretest, handed out first day of class (Mon Jan 11 2016) to be returned Wed Jan 13 2016.
- Convergence of Fourier Series (and some facts about density of spaces)

**Handouts from 5440:**

- Fixed point theorems
- Topology notes (2017), a slight revision of Topology notes (2014)
- Functional Analysis key theorems (10 pages, 2014), or Functional Analysis key theorems (2017 update, smaller font, 8 pages; updated Mon 10/23 and Fri 10/27)
- Review of sequences and series
- Orthonormal bases (no proofs), which includes a few theorems from Kreyszig. This is a bit simpler than in our book, since he treats unordered uncountable sums in a simpler way (using only their countably many nonzero terms).
- Orthonormal bases (with proofs) version

**Supplemental (non-required) texts/resources:**

The material we cover is quite standard, so there are lots of references. If you were to get one additional reference, I'd suggest Kreyszig.

- Royden's classic Real Analysis
- Kolmogorov and Fomin's Introductory Real Analysis, which is highly readable and not too long
- Kreyszig's Introductory Functional Analysis with Applications
- Kreyszig is a particularly good book for self-study because (1) he is rather conversational with good insight, (2) the topics are rather close to our syllabus, in that he covers the contraction mapping theorem, basic differential equations, and Fourier analysis, and (3) there are answers to odd-numbered exercises in the back, so you can check your work.

- There are not a lot of books on
*applied*functional analysis (meaning that the Fourier transform, distributions and Green's functions are discussed), and many of them have a physics bent. Applied functional analysis by D. H. Griffel is in similar spirit to our book and quite readable (and cheap on Amazon), so it comes recommended as a supplement. There is a book by J. T. Oden and L. F. Demkowicz with the same title which may also be useful, though not quite as cheap. - Applied Functional Analysis: Applications to Mathematical Physics by E. Zeidler is available on SpringerLink, so you have free access to the PDF version if you are on campus. I am not familiar with this book so no endorsement, but amazon reviews are positive.
- Elementary Functional Analysis by B. MacCluer, also available on SpringerLink (no endorsements)
- A course in Functional Analysis by John B. Conway (not Comway as Springer/Amazon say, nor the John H. Conway) goes into more detail in some places
- (2nd ed.1990, 1997 revised 4th printing)

- For a whirlwind tour of all of analysis (covers a lot, but not too deeply), see the Hitch-hiker's guide to analysis (2005), free download via SpringerLink if you are on campus
- Reed and Simon's Methods of Mathematical Physics, vol. 1: functional analysis, which is a bit condensed and focuses on some applications particularly relevant for applied mathematics
- Apostol's Mathematical Analysis covers real analysis and Fourier series
- For a modern, more conversational perspesctive, two good online sources from respected mathematicians are the following class notes:
- Alan Sokal's MATH 3103 Functional Analysis
- Terence Tao's 245B Real Analysis (the posts may not show up in a good order, but they are quite good and full of insight)
- get more ideas from posts like this at math.stackexchange
- some nice notes on 2nd semester graduate level real analysis from Christopher Heil at Georgia Tech (his 2008 course)

- Topics specific to the second semester (5450):
- For chapter 12 on measure theory, the classic is Measure and Integral by Wheeden and Zygmund, though this builds it up via Lebesgue measure first
- (you may also be able to find references on measure-theory from E. Stein and R. Shakarchi's "Real Analysis: Measure Theory, Integration, and Hilbert Spaces" 2005 )
- For basic Sobolev space information, the web is full of good summaries at the appropriate level, such as Ch. 2: Hilbert and Sobolev spaces and Fourier and Sobolev
- For discussion and intro to the usefulness of Fourier Series, see Terry Tao's introduction.

Note: you may **not** use theorems/lemmas/etc. from any of the above references in a homework problem without proof unless explicitly mentioned (with the exception of the Hunter/Nachtergaele book, from which you may use any stated fact from the current or previous chapters).

**Previous versions the 5450 course:**

- spring semester APPM 5450 (2015) and 2016, taught by Prof. Becker (links are probably dead as WebExpress automatically unpublishes old course webpages)
- spring semester APPM 5450 (taught by Prof Martinsson): 2006, 2007, 2008, 2010, 2011, 2013, 2014, 2017 (these links should still be active)

**Previous versions the 5440 course:**

- Taught by Prof. Becker in Fall 2014 (course webpage is gone, but material is similar to this semester)
- Taught by Prof. Martinsson in Fall 2016, Fall 2012, Fall 2009, Fall 2006 and Fall 2005.
- Prof. Li in 2013, and by Prof. Corcoran in 2010 and 2015.