# Matt N.

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 Jul3 comment Show that a space is separable. @DanielRust But the product space does not only consist of continuous maps. What am I missing? Jul2 comment Topology of Normed Space What Lost1 said but with $x \in X \setminus \{0\}$. Jul2 awarded Socratic Jul2 awarded Curious Jul2 awarded Inquisitive Jul1 revised Center of $GL_n(\mathbb R)$ is the set of matrices $\lambda I$ Tag added. Jul1 comment Center of $GL_n(\mathbb R)$ is the set of matrices $\lambda I$ @gniourf_gniourf Yeah. Except the question is not asking for a proof. It's asking for a proof-verification. Jun30 awarded Popular Question Jun28 comment The special orthogonal group is a manifold palio, shouldn't you correct the $n^2$ as pointed out to you in the comments? Jun28 revised The special orthogonal group is a manifold edited tags Jun28 comment Hahn Banach and separation of points Fast : ) And I kept re-reading your answer, wondering how this works when suddenly a wild edit appeared : ) Jun27 accepted Example of a set that is in $\mathbf V$ but not in $\mathbf L$ Jun26 comment Best Less-Famous Texts for Forcing Haim, note that Halbeisen does not contain any problems, neither solved nor unsolved. Jun24 revised Are simple functions dense in $L^\infty$? edited tags Jun24 comment Density of linear span of idempotents in $L^{\infty}$ @DanielFischer Could you tell me if I got it? I posted a tentative proof here. Jun24 comment Density of linear span of idempotents in $L^{\infty}$ @DanielFischer Oh, "essential range", I see (new word added to my vocabulary!). And now I also understand the second half of your comment. I thought we use compactness to cover the range using finitely many $\varepsilon$-balls. Then the inverse image of each ball yields a measurable set $S_k$. Pick $c_k$ to be any value in $f(S_k)$. If we choose $\varepsilon$ small enough it will make the error small enough. But boundedness is of course enough to do that. Jun24 comment Density of linear span of idempotents in $L^{\infty}$ @DanielFischer But why is the image of an (essentially) bounded function compact? I can only see that it's bounded (obviously). Jun24 comment Density of linear span of idempotents in $L^{\infty}$ @DanielFischer Oh, I see that you already gave this hint in your very first comment to this question! Thank you : ) Jun24 comment Density of linear span of idempotents in $L^{\infty}$ @DanielFischer Could you give me a hint on how to show that the simple functions are dense in $L^\infty$? What I have is that if $\varepsilon > 0$ then the goal is to find measurable sets $S_1,\dots, S_n$ and coefficients $c_1,\dots, c_n$ such that $$\|f- \sum_{k=1}^n c_k \chi_{S_k}\|_\infty < \varepsilon$$ Now I'm not sure how to actually construct the sets and determine the coefficients. Jun24 comment Density of linear span of idempotents in $L^{\infty}$ @DanielFischer Thank you! And: Yes, you're right (we are pedantic : ))