# Write a Review: An Introduction to Functional Analysis

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**2**reviews**2**completed

5 years, 3 months 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...
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.

**3**reviews**2**completed

4 years, 2 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 deta...
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.