A Theorem of Littlewood, Orlicz, and Grothendieck about Sums in L1(0,1)

Many are the delights and treasures to be found in Grothendieck’s Ž . famous or ought we say infamous? ‘‘Resume’’ 8 : his wondrous inequal ity, beautiful lifting theorems for continuous vector-valued functions, stunning characterizations of Hilbert spaces; the list goes on and on, with patience and grit the main necessities for the curious. One of our favorites has to do with series in L-spaces. Here’s what it Ž . 1 Ž . 2 says: if f is an unconditionally summable sequence in L , then f l n n n n 1 L . This is a crisp improvement of a theorem of Orlicz 16 which asserts 2 that Ý f and is the objective of this largely expository note. 1 n n To be sure, the L-space in question can be any L-space but for sake of 1Ž . clarity we will deal only with L 0, 1 .