We show that, for several function Banach spaces, multiplication considered as a bilinear continuous srjection, is an open mapping. In particular, we prove that multiplication from $L_p \times L_q$ to $L_1$ (for $p,q \in [1,\infty]$, $1/p + 1/q = 1$) is open.
"OPENNESS OF MULTIPLICATION IN SOME FUNCTION SPACES." Taiwanese J. Math. 17 (3) 1115 - 1126, 2013. https://doi.org/10.11650/tjm.17.2013.2521