$W$ is a $V$-module, then $D(W)=W'$.$\;\;\;\;\Box$

Suppose that $V$ contains a regular vertex operator subalgebra (with the same Virasoro element) $V^{0}$. Since any weak $V$-module is a weak $V^{0}$-module, $L(0)$ (the same for both $V$ and $V^{0}$) acts semisimply on any weak $V$-module. Then $V$ is of class $\cal{A}$, so that Proposition \[pdsub\] applies to $V$.

The weak $V\otimes V$-module ${\cal{D}}_{P(z)}(W)$
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In this section, given a weak $V$-module $W$ and a nonzero complex number $z$, we define a canonical subspace ${\cal{D}}_{P(z)}(W)$ of $W^{*}$, consisting of what we call $P(z)$-linear functionals on $W$. This space ${\cal{D}}_{P(z)}(W)$ contains $D(W)$ defined in Section 2 as a subspace. We prove that ${\cal{D}}_{P(z)}(W)$ has a natural weak module structure for the tensor product vertex operator algebra $V\otimes V$.

[*Throughout this section, $W$ is a weak $V$-module and $z$ is a nonzero complex number.* ]{} Now we introduce our first key notion.

All $P(z)$-linear functionals on $W$ clearly form a vector space, which we denote by ${\cal{D}}_{P(z)}(W)$.

Notice that a rational function in $x$ with only (two) possible poles at $0$ and $z$ can be considered as a meromorphic function with only three possible poles at $0, \infty$ and $z$ on the sphere $CP^{1}$. (Recall 