Boundedness of a sequence in Orlicz Space
0
$begingroup$
By the Definition of Orlicz function $M$, we know that $M(0)=0$, $M(x)>0$ for $x>0$ and $M(x)rightarrow infty$ as $xrightarrow infty$. And the Orlicz-Luxemborg norm is given by
$|x|=Inf{r>0:sum_{n=1}^{infty}M(frac{x_n}{r})leq 1}$
Using this, can we conclude that
$sum_{n=1}^{infty}M(frac{x_n}{r})leq 1$ implies $x_n$ is a bounded sequence.
sequences-and-series functional-analysis orlicz-spaces summability-theory
asked Dec 14 '18 at 16:33
J. YomchaJ. Yomcha
11
$endgroup$
|
show 3 more comments
0
$begingroup$
By the Definition of Orlicz function $M$, we know that $M(0)=0$, $M(x)>0$ for $x>0$ and $M(x)rightarrow infty$ as $xrightarrow infty$. And the Orlicz-Luxemborg norm is given by
$|x|=Inf{r>0:sum_{n=1}^{infty}M(frac{x_n}{r})leq 1}$
Using this, can we conclude that
$sum_{n=1}^{infty}M(frac{x_n}{r})leq 1$ implies $x_n$ is a bounded sequence.
sequences-and-series functional-analysis orlicz-spaces summability-theory
asked Dec 14 '18 at 16:33
J. YomchaJ. Yomcha
11
$endgroup$
$begingroup$
How is $x_n$ defined? What even is $x_n$?
$endgroup$
– SmileyCraft
Dec 14 '18 at 16:48
$begingroup$
$x_n$ is a sequence of non negative real numbers.
$endgroup$
– J. Yomcha
Dec 14 '18 at 16:52
$begingroup$
So for the definition of $|x|$ do you fix some $n$, or do you want $M(frac{x_n}r)leq1$ for all $n$?
$endgroup$
– SmileyCraft
Dec 14 '18 at 16:55
$begingroup$
I am confused by the definition of an Orlicz function, as the $x_n$ term comes out of nowhere and it is never mentioned when or how it is defined.
$endgroup$
– SmileyCraft
Dec 14 '18 at 17:03
$begingroup$
Edited... Please see now
$endgroup$
– J. Yomcha
Dec 14 '18 at 17:11
|
show 3 more comments
0
0
0
$begingroup$
By the Definition of Orlicz function $M$, we know that $M(0)=0$, $M(x)>0$ for $x>0$ and $M(x)rightarrow infty$ as $xrightarrow infty$. And the Orlicz-Luxemborg norm is given by
$|x|=Inf{r>0:sum_{n=1}^{infty}M(frac{x_n}{r})leq 1}$
Using this, can we conclude that
$sum_{n=1}^{infty}M(frac{x_n}{r})leq 1$ implies $x_n$ is a bounded sequence.
sequences-and-series functional-analysis orlicz-spaces summability-theory
asked Dec 14 '18 at 16:33
J. YomchaJ. Yomcha
11
$endgroup$
By the Definition of Orlicz function $M$, we know that $M(0)=0$, $M(x)>0$ for $x>0$ and $M(x)rightarrow infty$ as $xrightarrow infty$. And the Orlicz-Luxemborg norm is given by
$|x|=Inf{r>0:sum_{n=1}^{infty}M(frac{x_n}{r})leq 1}$
Using this, can we conclude that
$sum_{n=1}^{infty}M(frac{x_n}{r})leq 1$ implies $x_n$ is a bounded sequence.
sequences-and-series functional-analysis orlicz-spaces summability-theory
sequences-and-series functional-analysis orlicz-spaces summability-theory
asked Dec 14 '18 at 16:33
J. YomchaJ. Yomcha
11
asked Dec 14 '18 at 16:33
J. YomchaJ. Yomcha
11
edited Dec 14 '18 at 17:08
J. Yomcha
asked Dec 14 '18 at 16:33
J. YomchaJ. Yomcha
11
asked Dec 14 '18 at 16:33
J. YomchaJ. Yomcha
11
asked Dec 14 '18 at 16:33
J. YomchaJ. Yomcha
11
11
$begingroup$
How is $x_n$ defined? What even is $x_n$?
$endgroup$
– SmileyCraft
Dec 14 '18 at 16:48
$begingroup$
$x_n$ is a sequence of non negative real numbers.
$endgroup$
– J. Yomcha
Dec 14 '18 at 16:52
$begingroup$
So for the definition of $|x|$ do you fix some $n$, or do you want $M(frac{x_n}r)leq1$ for all $n$?
$endgroup$
– SmileyCraft
Dec 14 '18 at 16:55
$begingroup$
I am confused by the definition of an Orlicz function, as the $x_n$ term comes out of nowhere and it is never mentioned when or how it is defined.
$endgroup$
– SmileyCraft
Dec 14 '18 at 17:03
$begingroup$
Edited... Please see now
$endgroup$
– J. Yomcha
Dec 14 '18 at 17:11
|
show 3 more comments
$begingroup$
How is $x_n$ defined? What even is $x_n$?
$endgroup$
– SmileyCraft
Dec 14 '18 at 16:48
$begingroup$
$x_n$ is a sequence of non negative real numbers.
$endgroup$
– J. Yomcha
Dec 14 '18 at 16:52
$begingroup$
So for the definition of $|x|$ do you fix some $n$, or do you want $M(frac{x_n}r)leq1$ for all $n$?
$endgroup$
– SmileyCraft
Dec 14 '18 at 16:55
$begingroup$
I am confused by the definition of an Orlicz function, as the $x_n$ term comes out of nowhere and it is never mentioned when or how it is defined.
$endgroup$
– SmileyCraft
Dec 14 '18 at 17:03
$begingroup$
Edited... Please see now
$endgroup$
– J. Yomcha
Dec 14 '18 at 17:11
$begingroup$
How is $x_n$ defined? What even is $x_n$?
$endgroup$
– SmileyCraft
Dec 14 '18 at 16:48
$begingroup$
How is $x_n$ defined? What even is $x_n$?
$endgroup$
– SmileyCraft
Dec 14 '18 at 16:48
$begingroup$
$x_n$ is a sequence of non negative real numbers.
$endgroup$
– J. Yomcha
Dec 14 '18 at 16:52
$begingroup$
$x_n$ is a sequence of non negative real numbers.
$endgroup$
– J. Yomcha
Dec 14 '18 at 16:52
$begingroup$
So for the definition of $|x|$ do you fix some $n$, or do you want $M(frac{x_n}r)leq1$ for all $n$?
$endgroup$
– SmileyCraft
Dec 14 '18 at 16:55
$begingroup$
So for the definition of $|x|$ do you fix some $n$, or do you want $M(frac{x_n}r)leq1$ for all $n$?
$endgroup$
– SmileyCraft
Dec 14 '18 at 16:55
$begingroup$
I am confused by the definition of an Orlicz function, as the $x_n$ term comes out of nowhere and it is never mentioned when or how it is defined.
$endgroup$
– SmileyCraft
Dec 14 '18 at 17:03
$begingroup$
I am confused by the definition of an Orlicz function, as the $x_n$ term comes out of nowhere and it is never mentioned when or how it is defined.
$endgroup$
– SmileyCraft
Dec 14 '18 at 17:03
$begingroup$
Edited... Please see now
$endgroup$
– J. Yomcha
Dec 14 '18 at 17:11
$begingroup$
Edited... Please see now
$endgroup$
– J. Yomcha
Dec 14 '18 at 17:11
|
show 3 more comments
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$begingroup$
How is $x_n$ defined? What even is $x_n$?
$endgroup$
– SmileyCraft
Dec 14 '18 at 16:48
$begingroup$
$x_n$ is a sequence of non negative real numbers.
$endgroup$
– J. Yomcha
Dec 14 '18 at 16:52
$begingroup$
So for the definition of $|x|$ do you fix some $n$, or do you want $M(frac{x_n}r)leq1$ for all $n$?
$endgroup$
– SmileyCraft
Dec 14 '18 at 16:55
$begingroup$
I am confused by the definition of an Orlicz function, as the $x_n$ term comes out of nowhere and it is never mentioned when or how it is defined.
$endgroup$
– SmileyCraft
Dec 14 '18 at 17:03
$begingroup$
Edited... Please see now
$endgroup$
– J. Yomcha
Dec 14 '18 at 17:11