An Introduction to Functional Analysis

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An Introduction to Functional Analysis

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FREE

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Coursera online courses
Coursera's online classes are designed to help students achieve mastery over course material. Some of the best professors in the world - like neurobiology professor and author Peggy Mason from the University of Chicago, and computer science professor and Folding@Home director Vijay Pande - will supplement your knowledge through video lectures. They will also provide challenging assessments, interactive exercises during each lesson, and the opportunity to use a mobile app to keep up with yo...
Coursera's online classes are designed to help students achieve mastery over course material. Some of the best professors in the world - like neurobiology professor and author Peggy Mason from the University of Chicago, and computer science professor and Folding@Home director Vijay Pande - will supplement your knowledge through video lectures. They will also provide challenging assessments, interactive exercises during each lesson, and the opportunity to use a mobile app to keep up with your coursework. Coursera also partners with the US State Department to create “learning hubs” around the world. Students can get internet access, take courses, and participate in weekly in-person study groups to make learning even more collaborative. Begin your journey into the mysteries of the human brain by taking courses in neuroscience. Learn how to navigate the data infrastructures that multinational corporations use when you discover the world of data analysis. Follow one of Coursera’s “Skill Tracks”. Or try any one of its more than 560 available courses to help you achieve your academic and professional goals.

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Humanities
Sciences & Technology
4693 reviews

Course Description

Learn about functional analysis
Reviews 8/10 stars
8 Reviews for An Introduction to Functional Analysis

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Michael George profile image
Michael George profile image
10/10 starsCompleted
  • 2 reviews
  • 2 completed
5 years, 1 month ago
Having completed this course, I must comment that it is memorable, and I feel that despite its brevity, it is certainly one of the best courses I have ever taken. That said, one tends to derive from a course benefits proportional to the effort one puts in, and at least one other reviewer for this course, who wrote a rather short review, seemed to have dedicated essentially no effort, possibly because he or she has had the material at the freshman or sophomore level. That is, inherently, a rather superficial level. Furthermore, I do not feel that this course, overall, was taught at that level, although one might receive a passing grade of 60 out of 100 by approaching the course in this way. If that is all you want, don't read on. My review is not for you. But if you are interested in something of deeper value, this is what my review addresses. I have graduate degrees in mathematics and physics, and some prior experience studying funct... Having completed this course, I must comment that it is memorable, and I feel that despite its brevity, it is certainly one of the best courses I have ever taken. That said, one tends to derive from a course benefits proportional to the effort one puts in, and at least one other reviewer for this course, who wrote a rather short review, seemed to have dedicated essentially no effort, possibly because he or she has had the material at the freshman or sophomore level. That is, inherently, a rather superficial level. Furthermore, I do not feel that this course, overall, was taught at that level, although one might receive a passing grade of 60 out of 100 by approaching the course in this way. If that is all you want, don't read on. My review is not for you. But if you are interested in something of deeper value, this is what my review addresses. I have graduate degrees in mathematics and physics, and some prior experience studying functional analysis, but not in some of the areas in which this course focused. I found that this course compares favorably with other math courses I have taken at about the junior or senior level (or perhaps first-year graduate level). (There seems to be a lack of correspondence between European and American levels: This course seems to be at about a sophomore level for Europeans. It's a little hard to be certain. I teach math in the United States, and I must say it is not a sophomore level course, at least not for most students I encounter.) I took this course as a "nontraditional" student, who otherwise would not find courses of this quality available. This, as we all recognize, is the enormous advantage of MOOCs over traditional courses: Many people in the world now have access to high quality, sometimes very high quality, as in this course, courses, who otherwise would have few if any educational alternatives. The video lectures focused on developing intuition and insight. The fact that one can view the video lecture multiple times, and the fact that there is an active forum for the course makes a substantial difference over traditional courses, and I feel represents a possible improvement if one does not become a dilettante with respect to these MOOCs but really applies oneself seriously. However, my experience with online classes is that most students take the courses (often intentionally) as dilettantes, and fail to derive much. The video lectures were informative and insightful. The pdfs associated with the course went into more depth and detail, and helped to fill in some of the important material that could not be covered in the lectures. I found I also had to consult references. A very useful reference, at least for part of this course, is Berberian's text on Hilbert spaces. However, I also recommend that you have a good reference available for real analysis. Rudin's Principles of Mathematical Analysis seemed to be a favorite of many students. I used his text on real and complex analysis, which is a graduate level text. The quizzes were useful and helped sharpen my knowledge. I liked the fact that we could take most of the quizzes twice, as this allowed me to understand the mistakes I was making. Unfortunately, one can "game" many of the quizzes and obtain high scores that are unrepresentative of one's true level of achievement. The written homework (submitted for grading) involved proofs that provided some challenges. Writing proofs that will be graded by one's fellow students forces one to try to communicate as best one can. I definitely feel this was positive, as one often simply takes it for granted that a teacher or assistant will be able to decipher one's homework write-up, however obscure. In the end, after taking the final exam assessment, I could see that I was much more aware of whether or not I was communicating clearly. Overall, on these assessments, there seemed to be a lot of evidence for plagiarism despite severe penalties. Hopefully the future courses like this will introduce procedures to reduce this. The course had a lively discussion forum that was very interesting and worthwhile to contribute to. I was mostly interested in more "philosophical" issues than the pragmatics of most of the forum discussions. However, one could see that there was "room" enough for all of us, whether practical or theoretically inclined. This is quite a profound improvement over traditional courses. This alone makes MOOCs substantially of interest for students. It opens new territory that one does not ordinarily see in traditional courses. I found that the course subject matter lies close to my interests in physics and mathematics, and gave me a nice start for independent study and research. From applying myself seriously, I found that my level in this area improved significantly. Achieving this improvement, despite my good background in mathematics and physics, often meant working at the course beyond the 4 to 6 hours per week that is recommended in the description of the class. I typically spent 8 - 11 hours per week or more on this course. However, it did "pay off" in terms of learning as I made substantial progress. I also found that this course was able to capture some of the deep beauty of pure mathematics that makes us somewhat expect that this subject is applicable in physics and engineering. The last part of the course, and the very beginning portion, focus on applications to physics and math. As a "sophisticate", I have to say that the applications are quite impressive, although we in physics have our own approaches to what was addressed, the mathematical perspective was very elegant. I came to the course with pretty good preparation in mathematics, but I definitely think the course would be of value to a diverse student population. Certainly, the course is a serious time sink if one does not have a very good background, and the extent to which one could benefit is somewhat dubious, even if one tries when one does not have an adequate background. When I refer to "junior or senior level", as I did about this course previously, I meant as a major in pure mathematics. I disliked having to grade student homework, and each of us was required to grade at least five homework papers per week. This was too time-consuming for the limited time I had available each week to dedicate to the course. However, I do believe that evaluating homework was helpful in my own understanding. I was concerned that the course might prove to be too time-consuming overall as it is very intense for 8+ weeks. Also, the final exam was, albeit of moderate difficulty, quite long. However, I was able to manage time satisfactorily. My goal was to get well-enough prepared to study and apply more modern theory of functional analysis and partial differential equations, and for this purpose, this course was superb. I strongly recommend this course for people who work in an area sufficiently closely related to this area of analysis.
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4/10 starsCompleted
  • 3 reviews
  • 2 completed
3 years, 11 months ago
I was very excited to begin this course but, in the end, it was easily one of the worst classes I have taken through either edX or Coursera. The lectures covered the materials at an inadequate level of detail, such that reading the text was necessary to do truly well on the assessments. This had the effect of really transforming the class into one which was little more than reading the text and then taking an online assessment: it failed to leverage the power of the internet or multimedia environment. I did not take the final because it was, simply put, INSANE. It covered the material in a depth that was only appropriate to those who already had a deep background in mathematical theory and so did not remotely reflect the material of the course. All in all, a great disappointment.
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Mahmoud Fathy profile image
Mahmoud Fathy profile image
9/10 starsCompleted
  • 1 review
  • 1 completed
4 years, 5 months ago
Sadly I have only known about this course after it has been taken out I would have loved to attend this course, after reading about the course format , I have seen how it really suits what I precisely require in this subject. But it is taken out, I wish there is some way just to watch the videos , please :(
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leonard mangini profile image
leonard mangini profile image
10/10 starsCompleted
  • 39 reviews
  • 37 completed
5 years ago
Along with Exploring Quantum Mechanics, the most challenging course on Coursera, requiring dense weekly peer reviewed proofs. Requires significant "maturity" in terms of ability to take the basic quizzes and very in depth weekly readings and synthesizing to prove the classic proofs in the field. Incredibly well self-policed forum with more skilled students training the junior students in a rather dense field. I taught myself calculus at age 12 and found this course rather challenging with 10+ hours a week of required work to absorb the material
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Student

3/10 starsTaking Now
5 years, 1 month ago
An introductory course in linear analysis, similar in content to 2nd year undergrad courses in the UK. Lectures are clear and easy to follow. Unfortunately, the course is badly let down by the grading policy - much of which is based on peer assessment of proofs - often done by other students who openly admit in the forums they have no clue what's going on! So, OK for self study, but definitely do not pay for SigTrack!
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Hubert Chang profile image
Hubert Chang profile image
10/10 starsCompleted
  • 1 review
  • 1 completed
4 years, 11 months ago
While I was exposed to pure math courses before, I do not have the implicit pre-requisite for this course. However, given a determined mind plus encouragement from coursera peers, I finished the course. This is the first more advanced math course I took online. The video presentations from the lecturer are clear. The content is oriented toward solving PDE, which is good, as it makes the final objective clear and useful. There are also additional material in PDF format that are of more substances and context; it needs dedicated time to study and digest. Two forms of tests are provided: quizzes are helpful in clearifying misunderstood (or missed) concepts; peer review sets are helpful in sharping and tooling your understanding and problem solving ability. I learned a lot from the peer review sets as well as grading problems. The final is peer assessment problems and it was given fair enough time for students to finish. The online discu... While I was exposed to pure math courses before, I do not have the implicit pre-requisite for this course. However, given a determined mind plus encouragement from coursera peers, I finished the course. This is the first more advanced math course I took online. The video presentations from the lecturer are clear. The content is oriented toward solving PDE, which is good, as it makes the final objective clear and useful. There are also additional material in PDF format that are of more substances and context; it needs dedicated time to study and digest. Two forms of tests are provided: quizzes are helpful in clearifying misunderstood (or missed) concepts; peer review sets are helpful in sharping and tooling your understanding and problem solving ability. I learned a lot from the peer review sets as well as grading problems. The final is peer assessment problems and it was given fair enough time for students to finish. The online discussion forum is active and aspiring. Many enthusiastic students provides feedback. If you stick to the courses and do every homework, you will definitely gain more problem solving ability and knowledge. The course is likely to be easier if you have solid multi-variables calculus, linear algebra background (meaning, you got 3.5+/4.0 in grades.) and some point set topology. It will be easy, if on top of prescribed courses, you also have taken real analysis and advanced calculus with good grading. In this case, yes, you could probably spend only 4-6 hours/week. Otherwise, be determined and be prepared to spend major time, or else you can't follow through. This course is likely to be in US university senior level or at least, late junior level. I spend lots of time on it. Is it worthy? it is a definite and sounding Yes.
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Homer Thompson profile image
Homer Thompson profile image
10/10 starsCompleted
  • 9 reviews
  • 7 completed
5 years, 1 month ago
This is a quite difficult course, especially the final three weeks when you discuss the Lp spaces and their duals, then study distributions and Sobolev spaces, and finally end on PDEs and finite element approximation. I mean it's a real treat to see a course at this level taught for free in a very theoretical style (the so called French touch), and yet you end up touching briefly on a finite element method for solving PDE like it's done in the real world when you don't have simple closed form textbook examples. Really cool material all the way around. There is about an hour a week of lectures, which really aren't enough to understand the whole course. That's not to say they should be skipped. For instance, Prof Cagnol gives a very intuitive explanation of the Lebesgue integral as summing up horizontal slabs under a curve to contrast it to the Riemann integral that sums vertical columns under a curve. It drives me absolutely nuts that... This is a quite difficult course, especially the final three weeks when you discuss the Lp spaces and their duals, then study distributions and Sobolev spaces, and finally end on PDEs and finite element approximation. I mean it's a real treat to see a course at this level taught for free in a very theoretical style (the so called French touch), and yet you end up touching briefly on a finite element method for solving PDE like it's done in the real world when you don't have simple closed form textbook examples. Really cool material all the way around. There is about an hour a week of lectures, which really aren't enough to understand the whole course. That's not to say they should be skipped. For instance, Prof Cagnol gives a very intuitive explanation of the Lebesgue integral as summing up horizontal slabs under a curve to contrast it to the Riemann integral that sums vertical columns under a curve. It drives me absolutely nuts that you don't see this clear and illuminating diagram in some of the standard first year grad analysis books like Big Rudin or Royden, though Folland thankfully does seem to understand how much insight this provides when integrating simple functions. The lecture notes and the peer graded problems are the real meat of the course, and while the lectures give some great geometric intuition, they're pretty tough without working your way through the notes also. I think the prerequisites are understated. To succeed in the peer review problems a first real analysis course is a necessity. Functional analysis is way too advanced to be learning how to prove your first theorems. Some exposure to Lebesgue integrals is pretty important too. Not saying you need to have worked through Big Rudin or anything, but you do need to understand things like convergence almost everywhere and similar things relating to sets of measure zero. And 4-6 hours a week? No way if you're going for the advanced certificate (>= 80%). Probably not even if going for the 60% certificate, since almost 40% of the course grade is from Peer Review problems, which are all proofs. Overall, I feel like I learned a lot, even though I have a bachelor's in math with a couple of analysis courses under my belt (but no functional analysis before this course). It's a really enjoyable course if you like analysis, and the PDE capstone at the end really ties everything together beautifully. I really hope to see Prof Cagnol and Prof Rozanova-Pierrat give us some more French touch math on coursera, and hopefully they'll do this wonderful course again too. If you don't think this course is beautiful, you have no heart.
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Ian Wright profile image
Ian Wright profile image
10/10 starsTaking Now
  • 1 review
  • 0 completed
5 years, 1 month ago
This is a very fast paced course, and I wouldn't even think about taking it unless you have a strong mathematical background - probably at degree level with experience in analysis and point-set topology. It starts with the basics of convergence and continuity in topological and metric spaces, before moving on to concepts such as completeness, which lead to Banach spaces and Hilbert spaces. From there some applications and examples are discussed before moving onto Sobolev spaces and partial differential equations. All in 8 weeks. I spent about 3-4 hours per week on the course, but really only skimmed it and have come away with a superficial understanding of many of the concepts. I could easily have spent 10 times as much time on it in order to get a proper understanding. I'll certainly keep the notes for possible future reference. Overall I'd highly recommend doing the course if you have the appropriate background. The professor has a... This is a very fast paced course, and I wouldn't even think about taking it unless you have a strong mathematical background - probably at degree level with experience in analysis and point-set topology. It starts with the basics of convergence and continuity in topological and metric spaces, before moving on to concepts such as completeness, which lead to Banach spaces and Hilbert spaces. From there some applications and examples are discussed before moving onto Sobolev spaces and partial differential equations. All in 8 weeks. I spent about 3-4 hours per week on the course, but really only skimmed it and have come away with a superficial understanding of many of the concepts. I could easily have spent 10 times as much time on it in order to get a proper understanding. I'll certainly keep the notes for possible future reference. Overall I'd highly recommend doing the course if you have the appropriate background. The professor has a very engaging teaching style and the forums were very helpful and informative. I wouldn't hold too much sway with the final result/certificate however. I'm quite possibly going to end up with an 'advanced certificate', but there's no way that that describes my understanding of the material, which is sketchy at best.
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