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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 : ))
Jun
24
comment Density of linear span of idempotents in $L^{\infty}$
@DanielFischer Am I right that $f\in L^\infty$ is idempotent if and only if $f=\chi_S$ (characteristic function) for $S\subseteq \Omega$ measurable?
Jun
19
comment Coarsest and Finest Topology
Then it depends on whether the smallest generated topology is a subset of the given family or not. Maybe you could add the new assumption to the question and then someone more knowledgable than myself will write an answer.
Jun
19
comment Coarsest and Finest Topology
No. For example if the family does not contain the empty set.
Jun
18
comment Distribution of $\log X$
@t.b. There is a smoke signal for you. And a cave painting. Not sure you get pinged if I send you a comment but I'm going to try.
Jun
16
comment Another limit to evaluate: $ \lim\limits_{x \to \infty}\frac{x-\sin x }{x-\tan x}$
Copy pasting from the question into wolfram alpha gives $1$ for me.
Jun
14
comment “Every linear mapping on a finite dimensional space is continuous”
Isn't $T(E)$ of dimension $\le n$? (assuming $E$ is of dimension $n$)
Jun
13
comment In topology class, continuous and surjective problem
@topy It was a pleasure!
Jun
13
comment In topology class, continuous and surjective problem
@topy Regarding your question in the comment: If you endow $[a,b]$ with the subspace topology then $[a,y) = [a,b]\cap (-\infty,y)$ hence $[a,y)$ is open in $[a,b]$. It is not closed since if it was both closed and open we could write $[a,b]$ as a disjoint union of open sets which would contradict the fact that $[a,b]$ is connected.
Jun
13
comment In topology class, continuous and surjective problem
@topy They are open by the definition of the subspace topology: If $Y$ is any subset of $\mathbb R$ then a set $S\subseteq Y$ in the subspace topology on $Y$ is defined to be open if there exists an open set $O \subseteq \mathbb R$ such that $S= Y \cap O$. Now in your question $Y$ is $f([0,1])$. Since $(y,\infty)$ is open in $\mathbb R$ the set $f([0,1]) \cap (y,\infty)$ is open in $f([0,1])$ (in the subspace topology).
Jun
13
comment A problem about general topology.
@DanielFischer What I don't understand is: how can one check something is a base for the topology on $X$ when one is not given a topology on $X$?
Jun
7
comment Find the minimum distance that equal maximum inner product
Yes, I'm still trying to understand.
Jun
7
comment Find the minimum distance that equal maximum inner product
But left side of what?
Jun
7
comment Find the minimum distance that equal maximum inner product
Is there a condition missing or am I misunderstanding something?
Jun
7
comment Find the minimum distance that equal maximum inner product
Also, I'm not sure this holds. If $H=\mathbb R^2$ and $x$ is $(1,0)$, $M$ is the x-axis then the minimal distance between $x$ and the x-axis is zero but the minimal distance between points on the y-axis with $\|y\|=1$ is strictly greater than $0$.
Jun
7
comment Find the minimum distance that equal maximum inner product
What do you mean by "...LHS is $Px_0$ now how to show RHS is also $Px_0$"? I'm not being obtuse, I really don't understand. Could you elaborate a bit please?