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The set      L ω  = ⋂  p = 1 ∞  L p [ 0 , 1 ] {L_\omega } = \bigcap \nolimits _{p = 1}^\infty {{L_p}[0,1]}       is not equal to      L ∞  [ 0 , 1 ] {L_\infty }[0,1]       since      L ω  {L_\omega }       contains the function      −  ln ⁡  x - \ln x      . Using the theory of      L p {L_p}       spaces for finitely additive set functions developed by Leader [ 9 ] we will prove several necessary and sufficient conditions for the normability of a generalization of      L ω  {L_\omega }      . These include the equality and finite dimensionality of all the      L p {L_p}       spaces,      p ⩾  1 p \geqslant 1      .

On the normability of the intersection of 𝐿_{𝑝} spaces