Recent content by ELB27

Thanks! I indeed meant that ##x## and ##x'## are independent. The special case I entered into WolframAlpha is ##xy\delta (x-y) \overset{?}{=} x^2\delta (x-y)## to which it returned false. Following micromass' comment, I entered ##y=5## and it returned true as an "alternate form" while still...

Homework Statement I am trying to determine whether $$f(x)g(x')\delta (x-x') = f(x)g(x)\delta (x-x') = f(x')g(x')\delta(x-x')$$ where \delta(x-x') is the Dirac delta function and f,g are some arbitrary (reasonably nice?) functions. Homework Equations The defining equation of a delta function...

Thank you very much for your detailed answer! I really appreciate it! I will definitely refer to Velleman as you suggested (and kindly provided a link to). I don't have a particular application in mind for Zorich. Ideally I want a rigorous and comprehensive treatment of analysis with a slight...

Indeed, these things do go over my head, I am barely familiar with DeMorgan's law in its formal form - I will take the time to learn them. I already started going through the book of proof linked above - does it cover these subjects? It does cover logic in Section 2, is the level there sufficient?

Yes, I am familiar with these techniques but I never learned them in a specialized setting (learned them on the fly during self-study of linear algebra from LADW). Not sure what you mean by those "rules" but I have done some (perhaps semi-)formal proofs, again from LADW. Thank you very much...

Thank you all for your suggestions! I went through all the books and I intend to go over Halmos' Naive Set Theory and then move to Zorich's books which I find very impressive in terms of breadth and depth. If I will struggle, I will take more time to go through Abbott's text which, from what I...

Thank you for the suggestion! I will have a look. I picked up some set theory here and there (mainly the notations). Do you think I should devote some time exclusively to it? If so, can you recommend a good resource?

But what is the incentive for them to do so? If I understand the situation correctly, employers already get many more applications than they can handle and thus, every possible filter will only come in handy to them. People's ignorance/lack of motivation to push themselves to enter the working...

And yet this risk depends on the particular career path you choose and can be to a limited extent predicted. That's what planning is for. The checklist on the other hand is an efficient way to organize and execute those plans. Sure, to be truly successful, this checklist must be continuously...

By offering an alternative approach and proving that it works! Throughout this whole thread, people have repeatedly criticized various naive attitudes of students and tried to nail their causes, yet none offered any constructive alternative to those attitudes. A student must have some definite...

Thanks! I'll have a look. Concerning the Princeton Lectures, these seem to be 4 books on 4 different areas of analysis. In what order should I tackle them should I choose this series?

Hi everyone, I just finished self-studying linear algebra from Treil's Linear Algebra Done Wrong, and I want to continue with analysis. I don't know the requirements, so I'll just list what I know. I already know single and multi-variable calculus (to give a sense of my level, I knew everything...

Thanks again! I think I see it now. (I was referring to zeros as coefficients of basis vectors. ##\alpha_{j_k}^k = 0## for one or more ##j_k## in post #4. I should have been clearer - sorry for the confusion.) Thank you for the reply! I think you're referring to the basis I used in post #4 and...

Thank you very much! I was wondering however about whether or not only elements with zeros cannot be represented as a tensor product of vectors. I.e., if your proposed general case is the most general one or are there additional ones?

Yes, $$(1,0)\otimes (1,0) + (0,1)\otimes (0,1) = \sum_{j1,j2=1}^2 \beta_{j1,j2} b_{j1}^1\otimes b_{j2}^2$$ where ##b_1^{1,2}=(1,0), b_2^{1,2} = (0,1)##, ##b^k_{jk} \in V_k## and ##\beta_{1,1}=\beta_{2,2}=1 ; \beta_{1,2}=\beta_{2,1}=0##. To represent this functional as a tensor product of...