The definition of closure.
$begingroup$
Recently, i have seen the definition of closure as follows,
begin{equation}
text{cl}~C = bigcap_{varepsilon>0}(C + varepsilon B),
end{equation}
where $B$ is Euclidean unit ball: $B={x mid |x| leq 1}$.
I think it is right but i feel a little confused, because
begin{equation}
bigcap_{varepsilon>0} varepsilon B = lim_{varepsilon downarrow 0}varepsilon B = {boldsymbol{0}}.
end{equation}
Therefore, could i say
begin{equation}
begin{aligned}
text{cl}~C &= bigcap_{varepsilon > 0}(C + varepsilon B) \
&= lim_{varepsilon downarrow 0}(C + varepsilon B) \
&= C + lim_{varepsilon downarrow 0} varepsilon B \
&= C + {boldsymbol{0}}.
end{aligned}
end{equation}
I think it is incorrect but i don't know why, thanks in advance.
real-analysis linear-algebra convex-analysis
$endgroup$
add a comment |
$begingroup$
Recently, i have seen the definition of closure as follows,
begin{equation}
text{cl}~C = bigcap_{varepsilon>0}(C + varepsilon B),
end{equation}
where $B$ is Euclidean unit ball: $B={x mid |x| leq 1}$.
I think it is right but i feel a little confused, because
begin{equation}
bigcap_{varepsilon>0} varepsilon B = lim_{varepsilon downarrow 0}varepsilon B = {boldsymbol{0}}.
end{equation}
Therefore, could i say
begin{equation}
begin{aligned}
text{cl}~C &= bigcap_{varepsilon > 0}(C + varepsilon B) \
&= lim_{varepsilon downarrow 0}(C + varepsilon B) \
&= C + lim_{varepsilon downarrow 0} varepsilon B \
&= C + {boldsymbol{0}}.
end{aligned}
end{equation}
I think it is incorrect but i don't know why, thanks in advance.
real-analysis linear-algebra convex-analysis
$endgroup$
$begingroup$
What is the definition of $B$?
$endgroup$
– José Carlos Santos
Dec 18 '18 at 15:15
$begingroup$
$A+B$ means ${a+bmid ain A, bin B}.
$endgroup$
– Jiu
Dec 18 '18 at 15:16
$begingroup$
That doesn't answer my question.
$endgroup$
– José Carlos Santos
Dec 18 '18 at 15:17
$begingroup$
I am sorry for my carelessness and i have added the definition of $B$.
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 15:54
$begingroup$
@JoséCarlosSantos Acctually we better ask what the hell is the definition $lim_{varepsilon downarrow 0}(varepsilon B) $ ?
$endgroup$
– Red shoes
Dec 22 '18 at 0:29
add a comment |
$begingroup$
Recently, i have seen the definition of closure as follows,
begin{equation}
text{cl}~C = bigcap_{varepsilon>0}(C + varepsilon B),
end{equation}
where $B$ is Euclidean unit ball: $B={x mid |x| leq 1}$.
I think it is right but i feel a little confused, because
begin{equation}
bigcap_{varepsilon>0} varepsilon B = lim_{varepsilon downarrow 0}varepsilon B = {boldsymbol{0}}.
end{equation}
Therefore, could i say
begin{equation}
begin{aligned}
text{cl}~C &= bigcap_{varepsilon > 0}(C + varepsilon B) \
&= lim_{varepsilon downarrow 0}(C + varepsilon B) \
&= C + lim_{varepsilon downarrow 0} varepsilon B \
&= C + {boldsymbol{0}}.
end{aligned}
end{equation}
I think it is incorrect but i don't know why, thanks in advance.
real-analysis linear-algebra convex-analysis
$endgroup$
Recently, i have seen the definition of closure as follows,
begin{equation}
text{cl}~C = bigcap_{varepsilon>0}(C + varepsilon B),
end{equation}
where $B$ is Euclidean unit ball: $B={x mid |x| leq 1}$.
I think it is right but i feel a little confused, because
begin{equation}
bigcap_{varepsilon>0} varepsilon B = lim_{varepsilon downarrow 0}varepsilon B = {boldsymbol{0}}.
end{equation}
Therefore, could i say
begin{equation}
begin{aligned}
text{cl}~C &= bigcap_{varepsilon > 0}(C + varepsilon B) \
&= lim_{varepsilon downarrow 0}(C + varepsilon B) \
&= C + lim_{varepsilon downarrow 0} varepsilon B \
&= C + {boldsymbol{0}}.
end{aligned}
end{equation}
I think it is incorrect but i don't know why, thanks in advance.
real-analysis linear-algebra convex-analysis
real-analysis linear-algebra convex-analysis
edited Dec 18 '18 at 15:51
Ze-Nan Li
asked Dec 18 '18 at 15:09
Ze-Nan LiZe-Nan Li
286
286
$begingroup$
What is the definition of $B$?
$endgroup$
– José Carlos Santos
Dec 18 '18 at 15:15
$begingroup$
$A+B$ means ${a+bmid ain A, bin B}.
$endgroup$
– Jiu
Dec 18 '18 at 15:16
$begingroup$
That doesn't answer my question.
$endgroup$
– José Carlos Santos
Dec 18 '18 at 15:17
$begingroup$
I am sorry for my carelessness and i have added the definition of $B$.
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 15:54
$begingroup$
@JoséCarlosSantos Acctually we better ask what the hell is the definition $lim_{varepsilon downarrow 0}(varepsilon B) $ ?
$endgroup$
– Red shoes
Dec 22 '18 at 0:29
add a comment |
$begingroup$
What is the definition of $B$?
$endgroup$
– José Carlos Santos
Dec 18 '18 at 15:15
$begingroup$
$A+B$ means ${a+bmid ain A, bin B}.
$endgroup$
– Jiu
Dec 18 '18 at 15:16
$begingroup$
That doesn't answer my question.
$endgroup$
– José Carlos Santos
Dec 18 '18 at 15:17
$begingroup$
I am sorry for my carelessness and i have added the definition of $B$.
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 15:54
$begingroup$
@JoséCarlosSantos Acctually we better ask what the hell is the definition $lim_{varepsilon downarrow 0}(varepsilon B) $ ?
$endgroup$
– Red shoes
Dec 22 '18 at 0:29
$begingroup$
What is the definition of $B$?
$endgroup$
– José Carlos Santos
Dec 18 '18 at 15:15
$begingroup$
What is the definition of $B$?
$endgroup$
– José Carlos Santos
Dec 18 '18 at 15:15
$begingroup$
$A+B$ means ${a+bmid ain A, bin B}.
$endgroup$
– Jiu
Dec 18 '18 at 15:16
$begingroup$
$A+B$ means ${a+bmid ain A, bin B}.
$endgroup$
– Jiu
Dec 18 '18 at 15:16
$begingroup$
That doesn't answer my question.
$endgroup$
– José Carlos Santos
Dec 18 '18 at 15:17
$begingroup$
That doesn't answer my question.
$endgroup$
– José Carlos Santos
Dec 18 '18 at 15:17
$begingroup$
I am sorry for my carelessness and i have added the definition of $B$.
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 15:54
$begingroup$
I am sorry for my carelessness and i have added the definition of $B$.
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 15:54
$begingroup$
@JoséCarlosSantos Acctually we better ask what the hell is the definition $lim_{varepsilon downarrow 0}(varepsilon B) $ ?
$endgroup$
– Red shoes
Dec 22 '18 at 0:29
$begingroup$
@JoséCarlosSantos Acctually we better ask what the hell is the definition $lim_{varepsilon downarrow 0}(varepsilon B) $ ?
$endgroup$
– Red shoes
Dec 22 '18 at 0:29
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
The easiest way to analyze such a proof is by using a small example. Take $C=(-infty,0)$, so the closure is $(-infty,0]$. You say $lim_{varepsilon downarrow 0}((-infty,0) + varepsilon B) = (-infty,0) + lim_{varepsilon downarrow 0}(varepsilon B)$, which does not seem right.
$endgroup$
add a comment |
$begingroup$
This is the part where you made a mistake:
$$
lim_{varepsilon downarrow 0}(C + varepsilon B) = C + lim_{varepsilon downarrow 0} varepsilon B
$$
The operation of taking limit here is not distributive with respect to Minkowski sum.
$endgroup$
$begingroup$
I got it, thanks. By the way, i have never seen a closure could be written in such a limit ($lim(C+varepsilon B)$). Can I write like this?
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 15:59
1
$begingroup$
Yes, you can do that for a metric space. It essentially captures the fact that a closed set contains all of its limit points.
$endgroup$
– BigbearZzz
Dec 18 '18 at 16:06
add a comment |
$begingroup$
Writing an intersection $bigcap_{varepsilon>0}^infty A_varepsilon$ of sets satisfying $A_x subset A_y$ for $x<y$ as a limit is misleading to say the least. In this case it lead you to the false believe that this intersection is compatible with (Minkowski) sums.
That is, in general:
$$
bigcap_{varepsilon>0} (A_varepsilon+B_varepsilon) color{red}neq bigcap_{varepsilon>0} A_varepsilon + bigcap_{varepsilon>0} B_varepsilon.
$$
Consider the example where $C = (0,1)subset Bbb R$ and $B=[-1,1]$ is the closed unit ball. Note that
$$
bigcap_{varepsilon > 0} (C+varepsilon B) = bigcap_{varepsilon > 0} (-varepsilon, 1+varepsilon) = [0,1] = operatorname{cl},(0,1)
$$
while
$$
C + bigcap_{varepsilon > 0} varepsilon B = C + {0} = C = (0,1) color{red}neq operatorname{cl},(0,1).
$$
$endgroup$
$begingroup$
Thanks, i find that the $limit$ brings some misleading.
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 16:07
1
$begingroup$
Well, you can describe it as a kind of limit and this actually can be made more precise, but it behaves very different from the kind of point-wise limits you know from calculus. – Be careful when importing intuitions from other subjects that have related notations!
$endgroup$
– Christoph
Dec 18 '18 at 16:09
add a comment |
Your Answer
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The easiest way to analyze such a proof is by using a small example. Take $C=(-infty,0)$, so the closure is $(-infty,0]$. You say $lim_{varepsilon downarrow 0}((-infty,0) + varepsilon B) = (-infty,0) + lim_{varepsilon downarrow 0}(varepsilon B)$, which does not seem right.
$endgroup$
add a comment |
$begingroup$
The easiest way to analyze such a proof is by using a small example. Take $C=(-infty,0)$, so the closure is $(-infty,0]$. You say $lim_{varepsilon downarrow 0}((-infty,0) + varepsilon B) = (-infty,0) + lim_{varepsilon downarrow 0}(varepsilon B)$, which does not seem right.
$endgroup$
add a comment |
$begingroup$
The easiest way to analyze such a proof is by using a small example. Take $C=(-infty,0)$, so the closure is $(-infty,0]$. You say $lim_{varepsilon downarrow 0}((-infty,0) + varepsilon B) = (-infty,0) + lim_{varepsilon downarrow 0}(varepsilon B)$, which does not seem right.
$endgroup$
The easiest way to analyze such a proof is by using a small example. Take $C=(-infty,0)$, so the closure is $(-infty,0]$. You say $lim_{varepsilon downarrow 0}((-infty,0) + varepsilon B) = (-infty,0) + lim_{varepsilon downarrow 0}(varepsilon B)$, which does not seem right.
answered Dec 18 '18 at 15:16
LinAlgLinAlg
9,8791521
9,8791521
add a comment |
add a comment |
$begingroup$
This is the part where you made a mistake:
$$
lim_{varepsilon downarrow 0}(C + varepsilon B) = C + lim_{varepsilon downarrow 0} varepsilon B
$$
The operation of taking limit here is not distributive with respect to Minkowski sum.
$endgroup$
$begingroup$
I got it, thanks. By the way, i have never seen a closure could be written in such a limit ($lim(C+varepsilon B)$). Can I write like this?
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 15:59
1
$begingroup$
Yes, you can do that for a metric space. It essentially captures the fact that a closed set contains all of its limit points.
$endgroup$
– BigbearZzz
Dec 18 '18 at 16:06
add a comment |
$begingroup$
This is the part where you made a mistake:
$$
lim_{varepsilon downarrow 0}(C + varepsilon B) = C + lim_{varepsilon downarrow 0} varepsilon B
$$
The operation of taking limit here is not distributive with respect to Minkowski sum.
$endgroup$
$begingroup$
I got it, thanks. By the way, i have never seen a closure could be written in such a limit ($lim(C+varepsilon B)$). Can I write like this?
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 15:59
1
$begingroup$
Yes, you can do that for a metric space. It essentially captures the fact that a closed set contains all of its limit points.
$endgroup$
– BigbearZzz
Dec 18 '18 at 16:06
add a comment |
$begingroup$
This is the part where you made a mistake:
$$
lim_{varepsilon downarrow 0}(C + varepsilon B) = C + lim_{varepsilon downarrow 0} varepsilon B
$$
The operation of taking limit here is not distributive with respect to Minkowski sum.
$endgroup$
This is the part where you made a mistake:
$$
lim_{varepsilon downarrow 0}(C + varepsilon B) = C + lim_{varepsilon downarrow 0} varepsilon B
$$
The operation of taking limit here is not distributive with respect to Minkowski sum.
answered Dec 18 '18 at 15:28
BigbearZzzBigbearZzz
8,80921652
8,80921652
$begingroup$
I got it, thanks. By the way, i have never seen a closure could be written in such a limit ($lim(C+varepsilon B)$). Can I write like this?
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 15:59
1
$begingroup$
Yes, you can do that for a metric space. It essentially captures the fact that a closed set contains all of its limit points.
$endgroup$
– BigbearZzz
Dec 18 '18 at 16:06
add a comment |
$begingroup$
I got it, thanks. By the way, i have never seen a closure could be written in such a limit ($lim(C+varepsilon B)$). Can I write like this?
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 15:59
1
$begingroup$
Yes, you can do that for a metric space. It essentially captures the fact that a closed set contains all of its limit points.
$endgroup$
– BigbearZzz
Dec 18 '18 at 16:06
$begingroup$
I got it, thanks. By the way, i have never seen a closure could be written in such a limit ($lim(C+varepsilon B)$). Can I write like this?
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 15:59
$begingroup$
I got it, thanks. By the way, i have never seen a closure could be written in such a limit ($lim(C+varepsilon B)$). Can I write like this?
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 15:59
1
1
$begingroup$
Yes, you can do that for a metric space. It essentially captures the fact that a closed set contains all of its limit points.
$endgroup$
– BigbearZzz
Dec 18 '18 at 16:06
$begingroup$
Yes, you can do that for a metric space. It essentially captures the fact that a closed set contains all of its limit points.
$endgroup$
– BigbearZzz
Dec 18 '18 at 16:06
add a comment |
$begingroup$
Writing an intersection $bigcap_{varepsilon>0}^infty A_varepsilon$ of sets satisfying $A_x subset A_y$ for $x<y$ as a limit is misleading to say the least. In this case it lead you to the false believe that this intersection is compatible with (Minkowski) sums.
That is, in general:
$$
bigcap_{varepsilon>0} (A_varepsilon+B_varepsilon) color{red}neq bigcap_{varepsilon>0} A_varepsilon + bigcap_{varepsilon>0} B_varepsilon.
$$
Consider the example where $C = (0,1)subset Bbb R$ and $B=[-1,1]$ is the closed unit ball. Note that
$$
bigcap_{varepsilon > 0} (C+varepsilon B) = bigcap_{varepsilon > 0} (-varepsilon, 1+varepsilon) = [0,1] = operatorname{cl},(0,1)
$$
while
$$
C + bigcap_{varepsilon > 0} varepsilon B = C + {0} = C = (0,1) color{red}neq operatorname{cl},(0,1).
$$
$endgroup$
$begingroup$
Thanks, i find that the $limit$ brings some misleading.
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 16:07
1
$begingroup$
Well, you can describe it as a kind of limit and this actually can be made more precise, but it behaves very different from the kind of point-wise limits you know from calculus. – Be careful when importing intuitions from other subjects that have related notations!
$endgroup$
– Christoph
Dec 18 '18 at 16:09
add a comment |
$begingroup$
Writing an intersection $bigcap_{varepsilon>0}^infty A_varepsilon$ of sets satisfying $A_x subset A_y$ for $x<y$ as a limit is misleading to say the least. In this case it lead you to the false believe that this intersection is compatible with (Minkowski) sums.
That is, in general:
$$
bigcap_{varepsilon>0} (A_varepsilon+B_varepsilon) color{red}neq bigcap_{varepsilon>0} A_varepsilon + bigcap_{varepsilon>0} B_varepsilon.
$$
Consider the example where $C = (0,1)subset Bbb R$ and $B=[-1,1]$ is the closed unit ball. Note that
$$
bigcap_{varepsilon > 0} (C+varepsilon B) = bigcap_{varepsilon > 0} (-varepsilon, 1+varepsilon) = [0,1] = operatorname{cl},(0,1)
$$
while
$$
C + bigcap_{varepsilon > 0} varepsilon B = C + {0} = C = (0,1) color{red}neq operatorname{cl},(0,1).
$$
$endgroup$
$begingroup$
Thanks, i find that the $limit$ brings some misleading.
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 16:07
1
$begingroup$
Well, you can describe it as a kind of limit and this actually can be made more precise, but it behaves very different from the kind of point-wise limits you know from calculus. – Be careful when importing intuitions from other subjects that have related notations!
$endgroup$
– Christoph
Dec 18 '18 at 16:09
add a comment |
$begingroup$
Writing an intersection $bigcap_{varepsilon>0}^infty A_varepsilon$ of sets satisfying $A_x subset A_y$ for $x<y$ as a limit is misleading to say the least. In this case it lead you to the false believe that this intersection is compatible with (Minkowski) sums.
That is, in general:
$$
bigcap_{varepsilon>0} (A_varepsilon+B_varepsilon) color{red}neq bigcap_{varepsilon>0} A_varepsilon + bigcap_{varepsilon>0} B_varepsilon.
$$
Consider the example where $C = (0,1)subset Bbb R$ and $B=[-1,1]$ is the closed unit ball. Note that
$$
bigcap_{varepsilon > 0} (C+varepsilon B) = bigcap_{varepsilon > 0} (-varepsilon, 1+varepsilon) = [0,1] = operatorname{cl},(0,1)
$$
while
$$
C + bigcap_{varepsilon > 0} varepsilon B = C + {0} = C = (0,1) color{red}neq operatorname{cl},(0,1).
$$
$endgroup$
Writing an intersection $bigcap_{varepsilon>0}^infty A_varepsilon$ of sets satisfying $A_x subset A_y$ for $x<y$ as a limit is misleading to say the least. In this case it lead you to the false believe that this intersection is compatible with (Minkowski) sums.
That is, in general:
$$
bigcap_{varepsilon>0} (A_varepsilon+B_varepsilon) color{red}neq bigcap_{varepsilon>0} A_varepsilon + bigcap_{varepsilon>0} B_varepsilon.
$$
Consider the example where $C = (0,1)subset Bbb R$ and $B=[-1,1]$ is the closed unit ball. Note that
$$
bigcap_{varepsilon > 0} (C+varepsilon B) = bigcap_{varepsilon > 0} (-varepsilon, 1+varepsilon) = [0,1] = operatorname{cl},(0,1)
$$
while
$$
C + bigcap_{varepsilon > 0} varepsilon B = C + {0} = C = (0,1) color{red}neq operatorname{cl},(0,1).
$$
edited Dec 18 '18 at 16:05
answered Dec 18 '18 at 15:59
ChristophChristoph
12.4k1642
12.4k1642
$begingroup$
Thanks, i find that the $limit$ brings some misleading.
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 16:07
1
$begingroup$
Well, you can describe it as a kind of limit and this actually can be made more precise, but it behaves very different from the kind of point-wise limits you know from calculus. – Be careful when importing intuitions from other subjects that have related notations!
$endgroup$
– Christoph
Dec 18 '18 at 16:09
add a comment |
$begingroup$
Thanks, i find that the $limit$ brings some misleading.
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 16:07
1
$begingroup$
Well, you can describe it as a kind of limit and this actually can be made more precise, but it behaves very different from the kind of point-wise limits you know from calculus. – Be careful when importing intuitions from other subjects that have related notations!
$endgroup$
– Christoph
Dec 18 '18 at 16:09
$begingroup$
Thanks, i find that the $limit$ brings some misleading.
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 16:07
$begingroup$
Thanks, i find that the $limit$ brings some misleading.
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 16:07
1
1
$begingroup$
Well, you can describe it as a kind of limit and this actually can be made more precise, but it behaves very different from the kind of point-wise limits you know from calculus. – Be careful when importing intuitions from other subjects that have related notations!
$endgroup$
– Christoph
Dec 18 '18 at 16:09
$begingroup$
Well, you can describe it as a kind of limit and this actually can be made more precise, but it behaves very different from the kind of point-wise limits you know from calculus. – Be careful when importing intuitions from other subjects that have related notations!
$endgroup$
– Christoph
Dec 18 '18 at 16:09
add a comment |
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$begingroup$
What is the definition of $B$?
$endgroup$
– José Carlos Santos
Dec 18 '18 at 15:15
$begingroup$
$A+B$ means ${a+bmid ain A, bin B}.
$endgroup$
– Jiu
Dec 18 '18 at 15:16
$begingroup$
That doesn't answer my question.
$endgroup$
– José Carlos Santos
Dec 18 '18 at 15:17
$begingroup$
I am sorry for my carelessness and i have added the definition of $B$.
$endgroup$
– Ze-Nan Li
Dec 18 '18 at 15:54
$begingroup$
@JoséCarlosSantos Acctually we better ask what the hell is the definition $lim_{varepsilon downarrow 0}(varepsilon B) $ ?
$endgroup$
– Red shoes
Dec 22 '18 at 0:29