Study plan for Functional Analysis - Recommendations and critique

[MexChemE]

Hello, PF!

It’s been a while since I last posted. I am looking for a critique and recommendations regarding my study plan towards Functional Analysis and applications (convex optimization, optimal control), but first, some background:

- This plan is in preparation for my master’s thesis, I will be working on optimization techniques and I want to be as rigorous as possible.
- ChemE degree, and just beginning a MEng in Systems Engineering, might be able to take some graduate math electives.
- My strengths are a solid intuitive understanding of single variable calculus and transport phenomena modeling.
- Never had a formal course in linear algebra during undergrad, and I don’t remember much of my multivariable class. 5 years have passed since my graduation.
- No previous proof-writing experience.

Now, for the actual plan:

For the first phase I intend to study linear algebra and multivariable calculus. My options for LA are Applied Linear Algebra by Shores and Linear Algebra and its Applications by Strang. By what I’ve seen by skimming the table of contents, both books are quite similar in content although I think Shores includes more proofs and also more interesting exercises, but Strang is a classic, which one would you recommend? I also think it is interesting Strang includes a small section on Hilbert spaces.

For multivariable calculus I am set on Lang’s Calculus of Several Variables.

Then for the second phase I intent on studying a more advanced LA book and mathematical analysis. For this phase I am pretty much decided on Lax’s Linear Algebra and Its Applications. For analysis though, I think I am going for Apostol’s Mathematical Analysis, however, I also like the contents in Zorich’s two-volume book, but it is pretty long. What is your opinion on Apostol vs. Zorich?

After that I think I should be ready to tackle functional analysis. Axler’s graduate analysis book seems like a good choice, however, maybe it would be best to read a dedicated FA book such as Rudin, Kreyszig, Kolmogorov. Could you please share your thoughts on this?

I think on parallel with my study of FA, I could begin studying advanced optimization at the level of Luenberger, Rockafellar or Sasane.

The timeline I have for this is one year before starting the program, and then one and a half years before having to start to work on my thesis, so 2.5 years in total.

Thank you very much for your insights!

 

[WWGD]

I point to think about is that LA applies mostly to the finite-dimensional case, while FA deals mostly with infinite-dimensional ones.

 

[The Bill]

Strang is a good choice for your first introduction.

Lax looks like a good choice from the table of contents. I'd also recommend referencing Axler's Linear Algebra Done Right while working through Lax or whatever other text you choose. Axler has an abridged PDF of that on his website. That abridgement may suffice as an additional reference.

Whatever treatment of functional analysis you choose, I'd recommend also getting a copy of D.H. Griffel's Applied Functional Analysis as a companion. It's a medium-rigor applied text, and very readable. Used as additional reading material, it may help you get some perspective and intuition on some of the methods and concepts while working through your more rigorous text of choice.

 

[WWGD]

Strang is a good choice for your first introduction.

Lax looks like a good choice from the table of contents. I'd also recommend referencing Axler's Linear Algebra Done Right while working through Lax or whatever other text you choose. Axler has an abridged PDF of that on his website. That abridgement may suffice as an additional reference.

Whatever treatment of functional analysis you choose, I'd recommend also getting a copy of D.H. Griffel's Applied Functional Analysis as a companion. It's a medium-rigor applied text, and very readable. Used as additional reading material, it may help you get some perspective and intuition on some of the methods and concepts while working through your more rigorous text of choice.

IIRC, Griffen was free online at some point.

 

[MidgetDwarf]

Hello, PF!

It’s been a while since I last posted. I am looking for a critique and recommendations regarding my study plan towards Functional Analysis and applications (convex optimization, optimal control), but first, some background:

- This plan is in preparation for my master’s thesis, I will be working on optimization techniques and I want to be as rigorous as possible.
- ChemE degree, and just beginning a MEng in Systems Engineering, might be able to take some graduate math electives.
- My strengths are a solid intuitive understanding of single variable calculus and transport phenomena modeling.
- Never had a formal course in linear algebra during undergrad, and I don’t remember much of my multivariable class. 5 years have passed since my graduation.
- No previous proof-writing experience.

Now, for the actual plan:

For the first phase I intend to study linear algebra and multivariable calculus. My options for LA are Applied Linear Algebra by Shores and Linear Algebra and its Applications by Strang. By what I’ve seen by skimming the table of contents, both books are quite similar in content although I think Shores includes more proofs and also more interesting exercises, but Strang is a classic, which one would you recommend? I also think it is interesting Strang includes a small section on Hilbert spaces.

For multivariable calculus I am set on Lang’s Calculus of Several Variables.

Then for the second phase I intent on studying a more advanced LA book and mathematical analysis. For this phase I am pretty much decided on Lax’s Linear Algebra and Its Applications. For analysis though, I think I am going for Apostol’s Mathematical Analysis, however, I also like the contents in Zorich’s two-volume book, but it is pretty long. What is your opinion on Apostol vs. Zorich?

After that I think I should be ready to tackle functional analysis. Axler’s graduate analysis book seems like a good choice, however, maybe it would be best to read a dedicated FA book such as Rudin, Kreyszig, Kolmogorov. Could you please share your thoughts on this?

I think on parallel with my study of FA, I could begin studying advanced optimization at the level of Luenberger, Rockafellar or Sasane.

The timeline I have for this is one year before starting the program, and then one and a half years before having to start to work on my thesis, so 2.5 years in total.

Thank you very much for your insights!

Zorich covers pretty much similar topics as Apostol, but Zorich is gentler, spelling out more details. That is why it is larger. Since you did not have an intro linear algebra, and plan to start with an easy intro. This suggest you have not seen formal proof based mathematics? If so, those Analysis books you chose may be impenetrable. They are not bad books, but require some familiarity with proofs.

 

[MexChemE]

Zorich covers pretty much similar topics as Apostol, but Zorich is gentler, spelling out more details. That is why it is larger. Since you did not have an intro linear algebra, and plan to start with an easy intro. This suggest you have not seen formal proof based mathematics? If so, those Analysis books you chose may be impenetrable. They are not bad books, but require some familiarity with proofs.

Yes, I actually don’t have previous proof-writing experience, that is why I intend to start gently with Strang and Lang. If you don’t mind, what reference or book would you recommend to bridge the gap between Intro LA and multivariable calc towards analysis at the level of Apostol?

Thanks!

 

[MexChemE]

Strang is a good choice for your first introduction.

Lax looks like a good choice from the table of contents. I'd also recommend referencing Axler's Linear Algebra Done Right while working through Lax or whatever other text you choose. Axler has an abridged PDF of that on his website. That abridgement may suffice as an additional reference.

Whatever treatment of functional analysis you choose, I'd recommend also getting a copy of D.H. Griffel's Applied Functional Analysis as a companion. It's a medium-rigor applied text, and very readable. Used as additional reading material, it may help you get some perspective and intuition on some of the methods and concepts while working through your more rigorous text of choice.

Griffel’s contents look nice and very related to my interests. I will try to get a copy. Thanks!

 

[MexChemE]

I point to think about is that LA applies mostly to the finite-dimensional case, while FA deals mostly with infinite-dimensional ones.

I am aware, however, as I lack formal education in LA, I would like to get acquainted with the finite-dimensional case first. Besides, LA is also useful for optimization techniques.

 

[The Bill]

Yes, I actually don’t have previous proof-writing experience, that is why I intend to start gently with Strang and Lang. If you don’t mind, what reference or book would you recommend to bridge the gap between Intro LA and multivariable calc towards analysis at the level of Apostol?

Thanks!

Book of Proof by Hammack is good. He offers a free PDF of it on his faculty website: https://www.people.vcu.edu/~rhammack/BookOfProof/ There's a paperback version for about $22 if you prefer hardcopy.

Most discrete mathematics textbooks can fill this role, also. The one I know best is older editions of Discrete Mathematics and Its Applications by Kenneth Rosen. Used copies of the third edition hardcover, for example, can be found for under $10 shipped. These have the advantage of drawing many examples and applications from applied and theoretical computer science and several areas of mathematics.

But, on your timeline, it may pay to use a more focused proof-writing textbook like Hammack, Cummings, or Velleman. All three of those are popular for intro to proofs courses in North American undergraduate mathematics and theoretical computer science programs.

 

[MidgetDwarf]

Yes, I actually don’t have previous proof-writing experience, that is why I intend to start gently with Strang and Lang. If you don’t mind, what reference or book would you recommend to bridge the gap between Intro LA and multivariable calc towards analysis at the level of Apostol?

Thanks!

It would take about maybe 3 months, on average, going through the proofs book. Then another 3 months to a year working through analysis, on just R alone. Less/more time depending at the level you wish to learn it. Now you want to add linear algebra into the mix, so that's another 3 months at the minimum. This is taking into account the fact that you will be taking classes in Mech E. So, it looks like 2-3 years, and you should be done with your MS by then.

I think you should be more realistic in your goals. Ie., enroll into the mathematical proofs course. after that enroll in Analysis. Then proceed with self study. At least this way you will have a foundation, when you are trying to learn rigorous mathematics on your own. There are functional analysis books aimed at Physicist/Engineers who are not acquainted with pure mathematics.

From a mathematical perspective, the majority of students in the math department are introduced to Functional Analysis in Senior year of a strong BS program, but not requiring much measure theory ( or non at all, but it's there or some areas are skimped) . For everyone else, its typically a graduate level course which can be done before Measure Theory, or after. Typically after from what I have seen.

So you are essentially trying to learn the material that a beginning math graduate student starts with, without having any experience with proofs...

 

[MexChemE]

It would take about maybe 3 months, on average, going through the proofs book. Then another 3 months to a year working through analysis, on just R alone. Less/more time depending at the level you wish to learn it. Now you want to add linear algebra into the mix, so that's another 3 months at the minimum. This is taking into account the fact that you will be taking classes in Mech E. So, it looks like 2-3 years, and you should be done with your MS by then.

I think you should be more realistic in your goals. Ie., enroll into the mathematical proofs course. after that enroll in Analysis. Then proceed with self study. At least this way you will have a foundation, when you are trying to learn rigorous mathematics on your own. There are functional analysis books aimed at Physicist/Engineers who are not acquainted with pure mathematics.

From a mathematical perspective, the majority of students in the math department are introduced to Functional Analysis in Senior year of a strong BS program, but not requiring much measure theory ( or non at all, but it's there or some areas are skimped) . For everyone else, its typically a graduate level course which can be done before Measure Theory, or after. Typically after from what I have seen.

So you are essentially trying to learn the material that a beginning math graduate student starts with, without having any experience with proofs...

I realize my study plan is ambitious, so to speak. You do raise very valid points. So, let’s say I just brush up on LA and multivariable calc then jump into FA at the level of Griffel, Kreyszig or Sasane (the three books list just LA and calculus as prerequisites), what would I be missing going through that route instead of following a more rigorous path through real analysis?

Or, in different words, what would be the difference between an engineer learning FA from Kreyszig with just LA and calculus under their belt, versus a mathematician learning FA from Rudin’s text?

 

[MidgetDwarf]

I realize my study plan is ambitious, so to speak. You do raise very valid points. So, let’s say I just brush up on LA and multivariable calc then jump into FA at the level of Griffel, Kreyszig or Sasane (the three books list just LA and calculus as prerequisites), what would I be missing going through that route instead of following a more rigorous path through real analysis?

Or, in different words, what would be the difference between an engineer learning FA from Kreyszig with just LA and calculus under their belt, versus a mathematician learning FA from Rudin’s text?

Im not sure. I majored in Pure Mathematics, and a few units short of a physics BS. So I am unsure of how engineering majors learn this. I am currently taking functional analysis, and the way my course is structured, requires multivariable analysis, complex analysis, and some Fourier analysis. It is mainly proofs, with a few computations sprinkled throughout.

 

[MidgetDwarf]

I did not like Rudin POMA, so I decided not to look at his other books. The Komogorov book, is a graduate math book in Pure Mathematics. Readable, but not for someone who does not know simple Analysis.

 

[malawi_glenn]

These appears to be the easiest functional analysis books
An Introduction to Functional Analysis by Robinson
A Friendly Approach To Functional Analysis by Sasane
Linear Functional analysis by Rynne