Let $(X,d)$ be a metric space and $A$ be a subset of $X$. Then the following statements are equivalent. $A$ is nowhere dense. $\overline{A}$ doesn't contain any non-empty open set. Each non-empty open set has a non-empty open subset which is disjoint from $\overline{A}$ Each ...

I want to show uniform boundedness of the projector operator in Grand Lebesgue space. Note that, the norm of $f(x)$ in ${ L }^{ p) }\left( -\pi ;\pi \right) $ is $${ \left\| f \right\| }_{ p) }=\underset { 0<\varepsilon <p-1 }{ sup } { \left( \frac { \varepsilon }{ 2\pi } \int _{ -\pi }^{ \...