Hartshorne's Remark 7.8.2 This remark presents another form of proposition 7.3, in terms of linear systems. What troubles me is the second condition of this version. It's formulated as follows.

(2) $\mathfrak{d}$ separates tangent vectors, i.e., given a closed point $P\in X$ and a tangent vector $t\in T_P(X)=(\mathfrak{m}_P/\mathfrak{m}^2_P)'$, there is a $D\in\mathfrak{d}$ such that $P\in\mathrm{Supp}\ D$, but $t\notin T_P(D)$. Here we think of $D$ as a locally principal closed subscheme, in which case the Zariski tangent space $T_P(D)=(\mathfrak{m}_{P,D}/\mathfrak{m}^2_{P,D})'$ is naturally a subspace of $T_P(X)$.

There are many points that I really can't understand. First, some notations in his words are strange for me, e.g. what the prime symbol $'$ means in the tangent space. I have searched through all the pages of this book but I still can't find the related definition about this notation. Then I'm also confused about the associated closed subscheme of $D$. How to describe the $T_P(D)$? What is the $\mathfrak{m}_{P,D}$? Does it, I guess, equal to the maximal ideal of the local ring $\mathscr{O}_{P,X}/\mathscr{I}_{P,D}$, where $\mathscr{I}_{P,D}$ is the ideal sheaf of $D$? And overall, how to prove the equivalence between this and the original form of (7.3) ? How to relate these tangent spaces to the $k$-vector space $\mathfrak{m}_P\mathscr{L}_P/\mathfrak{m}_P^2\mathscr{L}_P$ in (7.3) ?
I hope someone could help explain this remark in detail. And thanks in advance.
 A: (7.3 $\implies$ 7.8.2 )
Suppose we are given a closed point $P \in X$ and a tangent vector $t \in T_P(X)$. Then, there exist $s \in V = \Gamma(X, \mathcal{L})$ by Proposition 7.7 such that $\phi(s_P) \in m_P$ and $t(\phi(\overline{s_P})) \neq 0$, where $\phi : \mathcal{L}_P \to \mathcal{O}_P$ is the obvious isomorphism. Then, let D = $(s)_0$. Then, as $\phi(\overline{s_P}) \not\in ker(t)$, we get that $t \not\in T_{P}(D)$. (Note that $m_{P,D} = m_P/(\phi(s_P))$. We have a surjective map $m_P/m_P^2 \to m_{P,D}/m_{P,D}^2$, which induces an injective map between the duals, $T_{P}(D) \to T_P(X)$. Note that all the maps in $T_P(D)$ have $\phi(\overline{s_P})$ in their kernel).
( 7.8.2 $\implies$ 7.3 ) We first fix a closed point $P \in X$. We know in general that for every subspace W of codimension 1 of $m_P/m_P^2$, there exist an element in the tangent space say $t$, such that $ker(t) = W$. By this obvious fact regarding any tangent space, and the above argument, we get that for every subspace W of $m_P/m_P^2$, there exist an element $s \in V$ such that $\phi(s_P) \in m_P$ and $\phi(\overline{s_P})\not\in W$. Hence, as $m_P/m_P^2$ is a finite dimesnional vector space, we get that $\{s \in V | \phi(s_P) \in m_P\}$ generate $m_P/m_P^2$.
A: Question:"I hope someone could help explain this remark in detail. And thanks in advance."
Answer: If $I \subseteq A$ is any ideal with $B:=A/I$ and $I \subseteq \mathfrak{p} \subseteq A$ is a prime ideal, you get an ideal $\mathfrak{q}:=\mathfrak{p}/I\mathfrak{p} \subseteq A/I$ and a diagram of maps
$$\require{AMScd}
\begin{CD}
A @>{p}>> B\\
@VVV @VVV \\
A_{\mathfrak{p}} @>{q}>> B_{\mathfrak{q}}
\end{CD}
.$$
Let $Y:=Spec(B) \subseteq X:=Spec(A)$.
You get an induced surjective map of cotangent spaces
$$q^*: \mathfrak{m}/\mathfrak{m}^2\rightarrow \mathfrak{n}/\mathfrak{n}^2.$$
When you dualize you get an induced injective map of tangent spaces
$$ i: T_{\mathfrak{q}}(Y) \rightarrow T_{\mathfrak{p}}(X).$$
The map $i$ is injective since $q^*$ is surjective. Since tangent and cotangent spaces are defined in terms of local rings and their maximal ideals, it follows that for any closed subscheme $Y \subseteq X$ and any point $y\in Y$, you get an inclusion of tangent spaces
$$T_y(Y) \subseteq T_y(X)$$
Example: If $S:=Supp(D) \subseteq X$ is a closed subscheme and $s\in S$ it follows there is a canonical inclusion of $\kappa(s)$-vector spaces
$$T_s(S) \subseteq T_s(X).$$
