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    Math REUs 2012

    I didn't know they'd respond this fast. I haven't heard back from any (including SUNY Potsdam, so I'm in the same boat as you guys). The deadline for Michigan is still two weeks away, and that's one of my top choices. I hope I'm in a sweet spot in the wait-list so they accept me, but not for...
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    Compactness and every infinite subset has a limit point

    If you try to prove limit point compactness is equivalent to sequential compactness, it's actually rather natural. I sketched the proof below, so don't read it if you want to figure it out for yourself. The main idea is that we can go back and forth between subsequences and infinite...
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    Find the fallacy in the derivative.

    Well f is only defined if x is a natural number, so it's not well defined as a function from R to R. Also, we can't apply the summation rule if the number of terms is changing.
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    Proof of lebesuge measurable function

    You should provide us some detail about what attempts you've made. From what you've given, I'm not sure what you're stuck on and how I can help. I'll assume you've tried the obvious thing, which is look at the definition, F is measurable if F^{-1}(a,\infty] is measurable. Since f is...
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    Math REU 2011

    I haven't heard back from anyone (except Cornell, who rejected me), so I suppose I'm not on any first-wave acceptances. To be sure, has anyone heard from OSU, Fairfield, Claremont, or CSUCI? Also, I'm not really sure what to do if I don't get into a math REU this summer. Does anyone have any...
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    If f(x) = 0 for every bounded linear map f, is x = 0?

    Well the identity function is a bounded linear map, so yes. If you mean bounded linear functional, this is still true by Hahn-Banach, because we can send x to ||x||, and send each vector of the form ax to a||x||, where a is in C. By Hahn-Banach, this extends to a bounded linear functional.
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    Find a basis for the space of 2x2 symmetric matrices

    The space of 2x2 matrices is in general isomorphic to a very familiar space. Think about the way addition of matrices and scalar multiplication work, and you should figure this out (and if you think about this for a while, you might realize a more general property about finite vector spaces...
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    P-adic volume of p-adic circle.

    I would guess you'd need measure and integration. I think Koblitz's P-adic numbers, p-adic analysis, and zeta-functions covers this. That said, are you sure this curve encloses a finite area?
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    Cauchy Sequences in General Topological Spaces

    "Cauchy" Sequences in General Topological Spaces Is there an equivalent of a Cauchy sequence in a general topological space? Most definitions I have seen of "sequence" in general topological spaces assume the sequence converges within the space, and say a sequence converges if for every...
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    Interested in Funtional Analysis

    I've studied some functional analysis at the level of Kolmogorov and Fomin, and that required a basic knowledge of linear algebra and real analysis. I also found this set of lectures online that might be useful if you're selfstudying or need another resource...