This is a great video! Planning to cover this series within a few days. It would of tremendous help if you (re)start a series like this again, where you do few selected problems from such core areas of math. These are actually fantastic learning resource for students like me who are learning math in isolation and doesn't have any resource person to talk to... You're like a friend from whom we can learn new stuff through discussion. Thank you for the video, today I learned about Convergence in measure, Chebyshev inq and Markov's inequality and a great problem from you!
I think the key for the not so obvious direction of the third problem is to observe that min(1, |f-f_n|) is a dominated sequence of measurable functions. So convergence in measure implies convergence in the L^1 norm.
I appreciated real analysis much more when I learned rigorous probability theory. It's similar ideas but somehow the change in perspective makes certain things more intuitive at least for me
Man ... I feel depressed about problem 2 ... this just feels so ... ugh ... so bullshit of a problem if you don't know the trick ... this just feels so lame, so contrived ...I tried solving it by adding and subtracting summands to complete the (f - 1)^3 and then I ended up with integral (f^3) dx = integral (f - 1)^3 dx + 10. I wish this led anywhere, but sadly it does not.
I don’t if this an odd request or not but would consider doing a video on your studying routine? Like how do you organize your daily life and manage between studying and reading?
The second problem seems a lot less intimidating if you substitute g = f - 1 into the given integrals from the start (which is natural since the condition f ≥ 1 becomes g ≥ 0 which is easier to work with). The integrals in the qn statement becomes ∫ g dx = 1, ∫ g^2 dx = 2 and ∫ g^3 dx ≥ 4. This makes it a lot easier to guess how to use cauchy-schwarz, since the numbers 1,2,4 are so much nicer than the given 2,5,14 You can also use cauchy-schwarz in the form ( ∫ (1 · g) g dx )^2 ≤ ( ∫ (1 · 1) g dx )( ∫ (g · g) g dx ) rather than ( ∫ g^1/2 · g^3/2 dx )^2 ≤ ( ∫ g^1/2 · g^1/2 dx )( ∫ g^3/2 · g^3/2 dx ), i.e. using the inner product =∫ (a · b) g dx rather than =∫ a · b dx Everything here is equivalent to the stuff in the vid, but it makes spotting the solution much more natural than trying to notice (f-1)^2=(f-1)^1/2(f-1)^3/2 and using cauchy directly
Totally agree. This is what I did. I also started by looking at Holder's inequality rather than Cauchy-Schwarz (but if you optimize that, you will just get C-S at the end of the day).
Hi, first of all I love your videos! I just finished my 1st semester of CS in Germany, but I've enjoyed the math modules (analysis and discrete structures) a lot more than the actual CS modules, and I am considering switching my major after the 2nd to math, as there won't be many more math classes to come and I won't be able to choose them later on either, but I am unsure if it is "worth" it, financially speaking - job prospects as a mathematician seem to mostly fall into research, which I don't think I would mind, but would probably not be that great salary-wise. I know that you have talked about the possibility of going into the financial sector after you finish your PhD, I assume as some sort of analyst? Although I still have ~6months to decide and a few (mostly CS with the exception of linear algebra) courses to attend before that, which I might find more interesting, I would still be interested in your thoughts about switching from CS to mathematics. You've said that you got your bachelor's in environmental science and then switched to math for graduate school, but unfortunately for me, I won't be able to do this, since the CS bachelor's doesn't contain the necessary modules to pursue a master's in mathematics, so I would actually have to switch to a math bachelor's. Either way I am interested to hear your opinion about this, and although the German and American system seem to be quite different, the general career paths should be the same. Also sorry for the long comment :)
If you genuinely love the subject of mathematics, then I encourage you to pursue it. Here in the US, I know that the job market is better for those that pursue CS and engineering, but I am unsure how it is in Germany. For me, there was no question that I wanted a PhD in math. So if you do decide to pursue mathematics, I would be absolutely sure before switching. I hope this helps :)
one thing to keep in mind is that the math courses are much harder than discrete structes and the other cs math courses. Not to discourage you, but i knew someone who switched for the same reason that you mention, but he then went back to cs noticing that you have to work all day for math to work out. There are interesting and difficult computer sciences classes as well, which feel like math, for example computability theory (Berechenbarkeit) oder Automatentheorie
Hi, I'm currently taking real analysis 1 in my undergraduate degree and I find it very useful to see some problems to know what to expect Thank you for showing them
Things like the second problem is why I always preferred Algebra. Analysis seems to rely on conjuring up a bunch of specific constructions to continue with a lot of the proofs that seem impossible to see unless you spend a lot of time on it or know it a priori.
I'm self studying this topic and this seems to be true to me. To me it seems that it's rigourously build up from the basic axioms and from definition to definition, so you really can't skip any step and have to understand how mathematics is build up from the gound up.
For the second proof you solved, I understand that the trick works; but how would I know when to use it? Is there a particular setup to look for? Can this problem be solved without using the trick? Is this really a trick or just a very creative way to rewrite a problem that isn't apparently obvious without doing a lot of work? Basically, I'm asking is the end result of all that work the trick itself without the tedious work written up in the proof? An example would be calculus textbooks not always including every step in getting from a to b. I’m asking because most professors I’ve had for classes want every step explicitly written out. Then I’ve had professors who have allowed students to use previously derived definitions, theorems, etc.,when writing proofs. Currently, I’m learning how to write proofs, and I’m have problems getting from one statement in a proof to the next statement. This is why I’m asking so many questions. By the way, my college majors are mathematics and statistics. Thanks
I’m not sure how to solve it without the trick. The author of the problem wanted us to use Cauchy Schwarz for the problem so it really was a test to see if we could identify it. The professor expects us to know how to identify and apply results from class.
I'm glad im not the only one who always doubts myself with triangle inequalities lol. Every time I really have to convince myself that my picture makes sense and that I haven't got it the wrong way around.
@@ridwanmulyana2199 I would say they are probably the same difficulty. Some IMO problems I look at and could probably do them but I would need a few weeks to prepare. They are different types of problems is the issue.
In problem 1, we are not assuming we are in the metric space of real numbers. In problem 2, there is no mention of convexity, so Jensen's inequality I do not think applies.
hey man love ur channel! Im a math major rn in complex analysis and im writing a research paper on the Mandelbrot set. do u have any good resources for that topic?
On the first proof he says one way to show that a set is closed is to show that the compliment is open now without using google explain where the intuition from this arises 1:22
