Is the Set {aX : a is in L_infinity} Necessarily Closed in L1 Spaces?

[DeanSkerl]

Im an applied statistician who hasnt looked at this stuff in years (And even then never took a class in it formally). So try and forgive me if my language is sloppy.

Consider the space of all d-dimensional bounded random vectors, L_infinity. Its Banach, hence complete and closed.

Consider a single fixed d-dimensional random vector in L1. Let's call it X.

Is the set

{aX : a is in L_infinity} necessarily closed? Here I am considering the dot product so that aX is a random variable.

Any help, references, etc would be greatly appreciated.


--Scott

 

[fresh_42]

This is an orbit of $X$ for the continuous left-multiplication of $L^\infty$, hence it is closed.