$\alpha(u,v,x):=\alpha(Y^{o}(u,x)v)$ converges to a rational function of $x$ with only two possible poles at $0$ and $z$, and we call such linear functionals $P(z)$-[*linear functionals*]{} on $V$ and denote by ${\cal{D}}_{P(z)}(V)$ the space of all $P(z)$-linear functionals. We define $Y^{R}(u,x)\alpha$ and $Y^{L}(u,x)\alpha$ by requiring $\< Y^{R}(u,x)\alpha,v\>$ to be the formal Laurent series expansion of the rational function $\alpha(u,v,x)$ in the domain $0<|x|<|z|$, and $\< Y^{L}(u,x)\alpha,v\>$ to be the Laurent series expansion of the rational function $\alpha(u,v,x+z)$ in the domain $0<|x|<|z|$. Our first main result is that $Y^{L}$ and $Y^{R}$ give rise to a structure of a weak $V\otimes V$-module on ${\cal{D}}_{P(z)}(V)$ in the sense that all the axioms defining the notion of a module except those involving the grading hold.

Vertex operator algebras are similar to Lie algebras. One of the similarities is that ideals are always two-sided due to the skew-symmetry. Consequently, a simple vertex operator algebra itself is an irreducible module. Most of the time, the adjoint module and other modules can be equally treated. Here, we construct a weak $V\otimes V$-module ${\cal{D}}_{P(z)}(W)$ for a general $V$-module $W$. Then, for any additional modules $W_{1}$ and $W_{2}$, we identify the space ${\cal{M}}[P(z)]^{W'}_{W_{1}W_{2}}$ of $P(z)$-intertwining maps of the indicated type 