It's a very short answer but this may be of interest to you:
Davis, Henry W.; Murray, F. J.; Weber, J. K. Jr., Families of $L_p$-spaces with inductive and projective topologies, Pac. J. Math. 34, 619-638 (1970). ZBL0187.05802.
https://projecteuclid.org/download/pdf_1/euclid.pjm/1102971942
