Showing inequalities for $l^p$ sequences

Question

If I show that an inequality (e.g. Holder or Minkowski) holds for the $L^p$ space, then can I automatically conclude that the inequality also holds for $\ell^p$ sequences, just by integrating wrt. the counting measure? Thanks!

Answer 1

If you didn't use any specific properties of the measure, then yes, since the sequence spaces are, as you note, also $L^p$ spaces. If you use specific properties of the measure, you need to check whether the counting measure has these properties. – Daniel Fischer