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Aug
15
revised How to prove that $\phi_n\uparrow f$, where $f\in R[a,b]$ and $\phi_n$ is like below?
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Aug
6
answered How to prove that $\phi_n\uparrow f$, where $f\in R[a,b]$ and $\phi_n$ is like below?
Aug
6
comment Show that $\mathcal{G}$ generates $\mathcal{B}_X$, the Borel $\sigma$-algebra of $X$.
Even more, $\mathcal{G}$ is a subset of $\tau$, and $\mathcal{B}_X=\sigma(\tau)$ is the smallest $\sigma$-algebra containing $\tau$, i.e., $\mathcal{G}\subseteq\tau\subseteq\sigma(\tau)=\mathcal{B}_X$, so $\mathcal{B}_X$ is a $\sigma$-algebra containing $\mathcal{G}$, from which follows that $\sigma(\mathcal{G})\subseteq\mathcal{B}_X$ (see first phrase of previous comment). This inclusion does not use the fact that $\mathcal{G}$ is a subbase.
Aug
6
comment Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball?
Of course, the set $U$ in my previous comment is star-shaped but not convex. In the convex case the procedure described here should work without too many problems.
Aug
6
comment Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball?
Simply scaling the rays does not give an homeomorphism in general. For example let $B$ be the open unit ball of $\mathbb{R}^2$, and let $U=B\cup\left\{(2x,2y):(x,y)\in B, y<0\right\}$. The distance from $0$ to the boundary is not continuous.
Aug
6
comment Show that $\mathcal{G}$ generates $\mathcal{B}_X$, the Borel $\sigma$-algebra of $X$.
By definition, the $\sigma$-algebra generated by a set $\mathcal{C}$ is the smallest sigma-algebra $\sigma(\mathcal{C})$ containing $\mathcal{C}$. You know that $\mathcal{B}_X$ is, by definition, generated by the topology, let's call it $\tau$, i.e., $\mathcal{B}_X=\sigma(\tau)$. If you show that $\tau\subseteq\sigma(\mathcal{G})$, you conclude that $\mathcal{B}_X\subseteq\sigma(\mathcal{G})$, by the first phrase. The reverse inclusion should be easy.
Aug
6
answered Equivalent Criteria for convergence of a sequence
Aug
6
reviewed Approve Has anyone heard of this maths formula and where can I find the proof to check my proof is correct? $\sum^n_{i = 1}i + \sum^{n-1}_{i=1}i = n^2$
Jul
31
revised Continuity of distance function
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Jul
24
revised Continuity of distance function
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Jul
24
answered Continuity of distance function
Jul
23
comment Can I always write a bounded operator $T$ as $T=R^{*}S$
If $H$ is separable, then every linear combination of such products $R^*S$ has separable image, and this is also true taking limits. Thus if $K$ is non-separable, the identity $id_K$ does not belong to the closure of the linear span of such products.
Jul
23
answered Prove the Supremum is attained.
Jul
23
answered Solving this n x n matrix equation with special structure
Jul
20
answered In a C*-algebra $A$, $x$ is self-adjoint iff $\lim_{t\to 0}(1/t)(\Vert 1-itx\Vert-1)=0$.
Jul
20
revised In a C*-algebra $A$, $x$ is self-adjoint iff $\lim_{t\to 0}(1/t)(\Vert 1-itx\Vert-1)=0$.
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Jul
16
accepted Relation between tracial states on von Neumann algebras and their GNS representations
Jul
16
answered Show that a subspace is closed in a Hilbert space $H$
Jul
16
asked Relation between tracial states on von Neumann algebras and their GNS representations
Jul
5
answered If a C*-algebra $A=\overline{\bigcup S}$, where $S$ is a class of prime C*-subalgebras, then $A$ is prime.