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Sometime after I started this last semester of my undergrad, I realized analysis was more important than I originally thought it was. I also realized I was on track to be very deficient in analysis knowledge; luckily, I have a very flexible degree program and can shore that up over the next two years.
Now, the point of this post:
To help me along this path I bought and have been working through Abbott's Understanding Analysis, I'm is about two weeks in and I am into the sixth chapter.
My question is, how well polished do my analysis skills really need to be? I don't have any idea since I'm working through it on my own. On Dr. Duffy's continuum of "get it working -> get it right -> get it optimized," I've gotten most everything working, and a decent portion I've got right, but pretty much all of it is far from optimized (of the stuff I've covered). Granted, it's been two weeks, I obviously didn't expect it to be done yet.
I'll use my Math Stat courses as an example, we used Engelhardt and Bain's 'Introduction to Probability and Mathematical Statistics' for the two course sequence (I think, at least we did for the second one). We would only be assigned 1/3-1/2 of the problems as recommended practice, and then the prof would draw on some other source for the HW. Of course, we worked through so many other problems in class that is made up for a lot of the discrepancy, and the course was more getting down main methods that can be applied to many different scenarios. By the end, I could answer almost all problems thrown at me straight up, but I'm honestly not sure if we just skimped on the really tricky problems or not or how she handled that. I suppose I could check. I'm not sure how it should work with this analysis content though, it's a different type of thing.
Should I treat this walkthrough like that, course, or do I need to have every proof presented/asked for polished to a T for future work and it needs to all be down pat? I'm not sure what the standard is. Analysis is more about learning all these concepts so you understand the foundations of mathematics and less about learning a format to figure out sufficient statistics or prove a distribution is a part of the REC. As an example, I know another member on here who said he went through the first four chapters and then just moved into stochastic calc and basic measure theory. Very different options out there.
Now, the point of this post:
To help me along this path I bought and have been working through Abbott's Understanding Analysis, I'm is about two weeks in and I am into the sixth chapter.
My question is, how well polished do my analysis skills really need to be? I don't have any idea since I'm working through it on my own. On Dr. Duffy's continuum of "get it working -> get it right -> get it optimized," I've gotten most everything working, and a decent portion I've got right, but pretty much all of it is far from optimized (of the stuff I've covered). Granted, it's been two weeks, I obviously didn't expect it to be done yet.
I'll use my Math Stat courses as an example, we used Engelhardt and Bain's 'Introduction to Probability and Mathematical Statistics' for the two course sequence (I think, at least we did for the second one). We would only be assigned 1/3-1/2 of the problems as recommended practice, and then the prof would draw on some other source for the HW. Of course, we worked through so many other problems in class that is made up for a lot of the discrepancy, and the course was more getting down main methods that can be applied to many different scenarios. By the end, I could answer almost all problems thrown at me straight up, but I'm honestly not sure if we just skimped on the really tricky problems or not or how she handled that. I suppose I could check. I'm not sure how it should work with this analysis content though, it's a different type of thing.
Should I treat this walkthrough like that, course, or do I need to have every proof presented/asked for polished to a T for future work and it needs to all be down pat? I'm not sure what the standard is. Analysis is more about learning all these concepts so you understand the foundations of mathematics and less about learning a format to figure out sufficient statistics or prove a distribution is a part of the REC. As an example, I know another member on here who said he went through the first four chapters and then just moved into stochastic calc and basic measure theory. Very different options out there.
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. I'll be working on understanding it anyway - but don't you worry I'll be staying/straying deeper in the coding/numerical methods/stats side as my focus. My analysis study is planned to be more of a consistent on-the-side thing right now just because I want to be good at it so there isn't a gap in my understanding.
