On the Maximum Partial Sum of Independent Random Variables.

this becomes false if (BI), (Be) and (B) are replaced by (Nt), (NM) and (N), respectively. This follows, even for p = 2 = q, from the above example proving that (NW) is not linear. Correspondingly, (Ne) cannot be interpreted as the dual space of (NP), since such an interpretation would involve the definition of a scalar product. 7. Let (Nt) denote the space which relates to the space (Nt) in the same way as the space (N2) relates to (N2). If p > 1 isarbitrary, the space (NO) is complete and linear. The proofs are the same as before. The situation can be summarized as follows: (BO) is a subspace of (NO) and (N-) is a subspace of (NP); all three of these spaces are complete; the first and third of them are linear but the second is not.