How does ${X}_{w}\subset {X^*}_{w}$ occur in modular metric space?

[ozkan12]

Let $d$ be a metric on $X$. Fix ${x}_{0}\in X$. Let ${d}_{\lambda}\left(x,y\right)=\frac{1}\lambda{}\left| x-y \right|$ and The two sets

${X}_{w}={X}_{w}\left({x}_{0}\right)=\left\{x\in X:{d}_{\lambda}\left(x,{x}_{0}\right)\to0\left( as \lambda\to\infty\right) \right \}$

and

${X^*}_{w}={X^*}_{w}\left({x}_{0}\right)=\left\{x\in X:\exists\lambda=\lambda\left(x\right)>0 such that {d}_{w}\left(x,{{x}_{0}}\right)<\infty\right\}$

Then, it is clear that ${X}_{w}\subset{X^*}_{w}$...How this happens ? Please can you explain ? Thank you for your attention...Best wishes...

 

[Siron]

Well, in fact the inclusion is quite intuitive. To prove it you could do the following. In stead of proving: $x \in X_w \Rightarrow x \in X_w^{*}$ you prove $x \notin X_w^{*} \Rightarrow x \notin X_w$. It will be straightforward then. Do you see?

 

[ozkan12]

Dear Siron,

First of all, thank you for your attention...But I couldn't prove this...Best wishes..

 

[Evgeny.Makarov]

Let $d$ be a metric on $X$. Fix ${x}_{0}\in X$. Let ${d}_{\lambda}\left(x,y\right)=\frac{1}\lambda{}\left| x-y \right|$

What is $|x-y|$? Is it the case that $X\subseteq\Bbb R$?

${X}_{w}={X}_{w}\left({x}_{0}\right)=\left\{x\in X:{d}_{\lambda}\left(x,{x}_{0}\right)\to0\left( as \lambda\to\infty\right) \right \}$

Doesn't this mean that $X_w=X$ since it is always the case that $\frac{1}{\lambda}d(x,y)\to 0$ as $\lambda\to\infty$?

${X^*}_{w}={X^*}_{w}\left({x}_{0}\right)=\left\{x\in X:\exists\lambda=\lambda\left(x\right)>0 such that {d}_{w}\left(x,{{x}_{0}}\right)<\infty\right\}$

What is $d_w$?

 

[ozkan12]

$X\subseteq R$ and ${d}_{w}$ is wrong, I wrote wrong it, it must be ${d}_{\lambda}$

 

[Evgeny.Makarov]

$X\subseteq R$

Then why do you write, "Let $d$ be a metric on $X$" and never use $d$ afterwards?

I believe both $X_w$ and $X_w^*$ are equal to $X$. For $X_w^*$ and $x\in X$, take $\lambda=1$; then $d_\lambda(x,x_0)=|x-x_0|<\infty$, so $x\in X_w^*$.

 

[ozkan12]

No paper '''arXiv:1112.5561v1.pdf...''' this link, there is something related to modular metric space...And in page 6 you will see definition 2.2...Please can you explain that how happened ${X}_{w}\subset {X^*}_{w}$ ? Thank you for your attention...Best wishes...