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Jul
3
comment Show that a space is separable.
@DanielRust But the product space does not only consist of continuous maps. What am I missing?
Jul
2
comment Topology of Normed Space
What Lost1 said but with $x \in X \setminus \{0\}$.
Jul
2
awarded  Socratic
Jul
2
awarded  Curious
Jul
2
awarded  Inquisitive
Jul
1
revised Center of $GL_n(\mathbb R)$ is the set of matrices $\lambda I$
Tag added.
Jul
1
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.
Jun
30
awarded  Popular Question
Jun
28
comment The special orthogonal group is a manifold
palio, shouldn't you correct the $n^2$ as pointed out to you in the comments?
Jun
28
revised The special orthogonal group is a manifold
edited tags
Jun
28
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 : )
Jun
27
accepted Example of a set that is in $\mathbf V$ but not in $\mathbf L$
Jun
26
comment Best Less-Famous Texts for Forcing
Haim, note that Halbeisen does not contain any problems, neither solved nor unsolved.
Jun
24
revised Are simple functions dense in $L^\infty$?
edited tags
Jun
24
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.
Jun
24
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.
Jun
24
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).
Jun
24
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 : )
Jun
24
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.
Jun
24
comment Density of linear span of idempotents in $L^{\infty}$
@DanielFischer Thank you! And: Yes, you're right (we are pedantic : ))