Harald Hanche-Olsen
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 Mar 30 comment Is $\mathbb{C}-\{0\}$ homeomorphic to $\mathbb{R}$? In this case, I think it's “trained minds think alike”. This argument is fairly standard, though perhaps not the first thing you think of if you haven't seen it before. Mar 30 answered Why is $\lim _{n \to \infty} n \arctan(\frac{x}{n})=x?$ Mar 30 answered Derivation of the polarization identities? Mar 30 reviewed Approve Does the following infinite series converge?? (Leibnitz) Mar 28 comment positive linear functionals are bounded in $C^*$-algebras @Freeze_S With notation as in the answer, $b=\frac12(a+a^*)$, so $\|b\|\le\|a\|$ by the triangle inequality. Similarly, $\|c\|\le\|a\|$. So $|f(a)|^2=|f(b)+if(c)|^2=f(b)^2+f(c)^2\le C^2(\|b\|^2+\|c\|^2)\le 2C^2\|a\|^2$, where $C=f(e)$. Mar 27 comment Show that if $w^3=1$, then $1+w+w^2=0$ Yes, but it should have been stated explicitly. Mar 27 comment Correct typography for using Leibniz Notation How can one possibly answer a question like the $\iota$ one? There are a lot of letters that could be used, but aren't. Yes, it's an ISO standard, though when I last saw it years and years ago, it was as an NS (Norwegian Standard, which follows the ISO standards closely). Mar 27 comment Difference between $\ell ^{\infty}$ and $\ell^p$ As @Wouter said. However, note that this is for finite sequences. Still, it serves to motivate the definition of $\ell^\infty$ even for infinite sequences (just replace the maximum by a supremum). Mar 27 comment Correct typography for using Leibniz Notation The vast majority of mathematics journals and books use italic $d$, I think. But there exists an international standard for notation in technical and engineering fields that specifies the upright $\mathrm{d}$ for this purpose, and similarly for $\mathrm{i}=\sqrt{-1}$ and $\mathrm{e}=2.71828…$ (sorry, I don't have the reference on hand). Mar 25 comment For differentiable functions $f,g$, $\nabla f(x)=g(x)x$. Then $f$ is constant on S. To add some intuition to what @AndrewD.Hwang said: The assumption says that $\nabla f$ points in the radial direction, which means that $f$ will not change when you move along a sphere centered at the origin. So $f$ should be a function of $\|x\|$ alone. Mar 20 comment series divergence $\sum_{k=1}^{\infty}\sqrt{\tan^{-1}(\frac{1}{k^2})}$ Actually, if you examine the proof of the validity of the limit comparison test, you will see that it does produce a series of smaller terms to compare with. Mar 19 comment Is there a definition of ${\forall}$ that doesn't use the concept of propositional function? A similar comment applies to the concept of a propositional function, which is usually considered a meta concept; i.e., it is not a concept within the theory under study, but rather a concept about this theory. Mar 19 comment Is there a definition of ${\forall}$ that doesn't use the concept of propositional function? I have never come across a formal system in which $\forall$ is subject to definition. In formal first-order logic, the symbol is just there, and has to satisfy the axioms of first-order logic. There is, however, a definition of the interpretation of $\forall$ in a model in first order logic, and that definition resembles the one you gave. However, note that this definition is not a definition inside the first-order logic, so there is no circularity involved. Feb 25 revised How to write Kelvin equation in a different way Fix some LaTeX Feb 25 comment How to write Kelvin equation in a different way You are aware that $\ln(\mathrm{e}^x)=x$, right? (Some people like to write $\mathrm{e}=2.71828…$ with an upright e. When you have other $e$s in the same context, this becomes essential.) Feb 11 comment Triangles - sin, cos etc. @Nehorai In many parts of the world (e.g., much of Europe) one uses the comma instead of a period as decimal separator. Feb 11 comment What is the sum of all the natural numbers between $500$ and $1000$. Looks like a good one to emulate, yes. How about: Find the sum of all even numbers in the given range. Find the sum of all multiples of 14 in the given range. Subtract. Feb 5 answered How to prove $\lim_{s \rightarrow \infty} \zeta(s) = 1$? Feb 2 comment Idea behind the tangential vector space? To expand a little bit on the comment by @HagenvonEitzen, try to replace $U$ by the unit sphere in $\mathbb{R}^n$. You will find that the tangent space at any point is $n-1$-dimensional, and can in fact be identified with the hyperplane touching the sphere at the given point. And yet, there is no sensible way to identify all these spaces with $\mathbb{R}^{n-1}$ in a coherent way. Jan 31 comment Banach Tarski proof understanding Probably, assuming that $S$ being decomposable into the elements of $X$ means that you can write $S=P_1\cup\cdots\cup P_n$ (with the $P_i$ pairwise disjoint) and $P_i$ congruent to $Q_i$ for $i=1,\ldots,n$.