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 Apr 7 comment Compact operators are orthogonally equivalent to a diagonal matrix? @user1952009 I need to point out that it's wrong without restriction: consider the case $T$ has infinite singular values but $0$ is also a singular value (namely, $\ker T\neq0$). Apr 7 comment Compact operators are orthogonally equivalent to a diagonal matrix? @user1952009 Then, for example, why does $(e_n)_n$ constitutes a Hilbert base? I mean, it (Hilbertly) spans $H$. Apr 7 comment Compact operators are orthogonally equivalent to a diagonal matrix? @user1952009 The real issue is "at" the singular value $0$. In fact, even in general, for polar decomposition, we need to assume partial isometric instead of unitary because of $0$. If $\ker T$ is isometric to $\ker T^*$, then everything works for unitary operators in polar decomposition. Apr 7 asked Compact operators are orthogonally equivalent to a diagonal matrix? Mar 23 revised Weak-* bounded, closed convex set is compact? added 164 characters in body Mar 23 comment Weak-* bounded, closed convex set is compact? @user1952009 Thanks, you are right. It works. I remembered the general theorem that weak boundedness is equivalent to original boundedness for locally convex spaces, and if $E$ is reflexive, weak-* topology coincides with weak topology. Mar 23 asked Weak-* bounded, closed convex set is compact? Mar 18 comment Does the Closed Graph Theorem follow from Banach-Steinhaus? I don't think they are equivalent in some sense. At least, consider how strong the axiom of choice is needed for these theorems. See, for example, this paper and here. Mar 16 awarded Nice Question Mar 14 revised The set of curves of degree $d$ with two singular points or a degenerate singular point is closed? added 24 characters in body Mar 14 comment Use kruskal's algorithm to show that if G is a connected graph, then any subgraph that contains no circuits is part of some spanning tree for G. Hint: Given the graph $G=(V,E)$ and a subset $S\subseteq E$ without cycle. Consider the weight function $w=\chi_{E\setminus S}\colon E\to\mathbb\{0,1\}$ such that $w(e)=0$ for $e\in S$ and $w(e)=1$ for $e\in E\setminus S$. Mar 14 revised The set of curves of degree $d$ with two singular points or a degenerate singular point is closed? deleted 10 characters in body Mar 14 asked The set of curves of degree $d$ with two singular points or a degenerate singular point is closed? Mar 8 comment Smooth surfaces that isn't the zero-set of $f(x,y,z)$ @AmitaiYuval The existence of open covering of $\mathbb R^n$, but not $S$. Mar 6 comment Is topology on $C[0,1]$ metrizable? @XiangYu $C[0,1]$ is a non-closed proper subspace of $\mathbb R^{[0,1]}$. Mar 5 revised Show that in $f:A\to B$, and $Z\subset B$, we have: $f(f^{-1}(Z))\subset Z$ edited tags Mar 4 accepted On the Banach–Alaoglu theorem: is the unit ball of an equivalent norm also weak-* compact? Mar 1 comment On the Banach–Alaoglu theorem: is the unit ball of an equivalent norm also weak-* compact? Unless the nontrivial thing mentioned in the answer, everything seems right to me (wait for reviews from others). Facts used: 1. The dual of $X^*$, with respect to weak-* topology, is $X$ itself; 2. Suppose $T\colon X\to Y$ is continuous linear between TVSs, then the transpose ${}^tT$ is well-defined and weak-*$\to$weak-* continuous; 3. A linear map between normed spaces $T\colon E\to F$ is continuous if and only if it's weak-weak continuous. Mar 1 revised On the Banach–Alaoglu theorem: is the unit ball of an equivalent norm also weak-* compact? added 1 character in body Mar 1 revised On the Banach–Alaoglu theorem: is the unit ball of an equivalent norm also weak-* compact? added 21 characters in body