  In this paper we present (quasi-)norm equivalence on a vector-valued function
space $L^p_A(l^q)$ and extend the equivalence to $p=\infty$ and $0<q<\infty$ in
the scale of Triebel-Lizorkin space, motivated by Fraizer-Jawerth. By applying
the results, we improve the multilinear Hormander's multiplier theorem of
Tomita, that of Grafakos-Si, and the boundedness results for bilinear
pseudo-differential operators, given by Koezuka-Tomita.
