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 Apr 17 comment Exercise 4 chapter 7 Evans book Great proof! Do you happen to have a reference for the theory of Bochner spaces? Apr 14 comment Cauchy-Schwarz inequality for dual pairing? Ah, pretty obvious. Thanks! Apr 8 comment Prove the theorem of row swapping determinants?? See Freidberg, Insel, and Spence Linear Algebra. Feb 19 comment Find $\nabla\cdot (\frac{x}{|x|})$ Great. Is this because we regard $|x|$ as a scalar function? Nov 25 comment Proving harmonic function is zero I think you meant the boundary of omega. Hint: use a connectivity argument. Nov 25 comment Exchanging Series with Integrals: when is it possible? Note that by linearity, it's always possible to do this for finite sums. For infinite sums, it becomes a matter of exchanging a limit and an integral which you can use Fubini's convergence theorem, monotone convergence theorem, Vitali's convergence theorem, etc. Nov 25 comment Convergence of $\sum_{-\infty}^{\infty}e^{-\pi tn^2}$ Thank you very much. I had an inkling that it was this simple. Jul 6 comment Prove that $V$ is the direct sum of $W_1, W_2 ,\dots , W_k$ if and only if $\dim(V) = \sum_{i=1}^k \dim W_i$ I think you mean the sum of the dimension of the span of $B_i$ is the dimension of $W_i$. Jul 4 comment Let $f=g$ on $[a,b]/E$ where $f\in \mathcal{R}[a,b]$ and continuous on $[a,b]$. Then $g\in\mathcal{R}[a,b]$ and $\displaystyle\int_a^b f=\int_a^bg$. My question would be if this proof is correct. I saw this problem in an old analysis book and so I thought I would try it out. Sorry, I should have been more clear. Jul 4 comment Let $f=g$ on $[a,b]/E$ where $f\in \mathcal{R}[a,b]$ and continuous on $[a,b]$. Then $g\in\mathcal{R}[a,b]$ and $\displaystyle\int_a^b f=\int_a^bg$. Sorry about that, I have corrected the statement in my edit. May 27 comment $\lim_{n\to \infty} n^{1/n^2}$ Ahh, this is much better. Thanks! May 26 comment Bounded sequence of positive numbers Do you mean $r$ instead of $K$? Apr 14 comment Improper Integral with trigonometric functions Well this integral diverges so my integral will diverge by the comparison test. Correct? Mar 17 comment Residue Theorem and Homologous to zero In your definition of being homologous to zero, if there is a point outside of $G$ that isn't so that the winding number is zero, then wouldn't that mean the the curve does not meet the hypothesis of the RT? Mar 17 comment Residue Theorem and Homologous to zero This is the heart of my question I think. If $C_2$ is NOT homologous to 0, then it does not satisfy the hypothesis because in order to use the Residue Theorem, we need a curve that is homologous to 0. I apologize if I'm being stupid here... Mar 17 comment Residue Theorem and Homologous to zero So this was my question; we don't consider such curves as $C_2$ for the residue theorem because it does not satisfy the hypothesis. Correct? Mar 17 comment Residue Theorem and Homologous to zero Well in this picture the answer is no, you can't deform $C_2$ to a single point and still stay in $K$. Mar 17 comment Residue Theorem and Homologous to zero So in this picture, $C_2$ is not homologous to 0? Then for the Residue Theorem we couldn't use a curve such as $C_2$ but can only consider curves that are either $C_1$ or curves that wrap around $H$ in such a way so that the winding number is 0, correct? Mar 17 comment Residue Theorem and Homologous to zero I think I'm a little confused here. In this example $f=\frac{1}{z^2}$ and so is analytic on $\mathbb{C}-\{0\}$. Then if our contour is just the unit circle traversed once around this point, shouldn't the winding number be 1? Feb 19 comment Consider the function I would try a sequential argument and use the denseness of $\mathbb{R}$.