Dear George, Thank you for your interest. Yes, it is ()_j and same for ()_j  ( is the inner product).

Thank you Dear George.

Dear Peter Breuer,

Thank you for your response. About the norm, yes it's the uniform one, but it is also about random variables.

Dear Peter,

You are right, one can consider the boundedness of the random variable as its values belong to a bounded set (hypercube). Thank you