Suppose $X$ is a $k$-scheme and $\mathscr{L}$ an invertible sheaf on $X$. Show that a (finite-dimensional) base-point-free linear series $V$ on $X$ correspondoing to $\mathscr{L}$ induces a morphism to projective space
$$ \phi_V: X \to \Bbb{P}V^{\vee} $$. Does this result hold for any dimensional or only finite-dimensional?
It's Exercise 15.2.J(a) in Vakil's FOAG, page 429.
According to Vakil's definition in page 426, a linear series, abbreviated as $V$, is a triple of data $$(V, \mathscr{L}, \lambda : V \to \Gamma(X, \mathscr{L}))$$ with $V$ a vector space on $k$ and $\mathscr{L}$ an invertible sheaf on $X$ and a linear map $\lambda: V \to \Gamma(X, \mathscr{L}))$. Being base-point-free means that for all $v \in V$, $\lambda(v)$ has no common zeros:
$$ \bigcap_{v \in V} \text{vanish set of }\lambda (v) = \varnothing $$
Whereas the definition of projective space Vakil taking is the $\operatorname{Proj}$ of the symmetric algebra of the dual vector space, hence $\Bbb{P}(V^{\vee}) = \operatorname{Proj}\operatorname{Sym}V^{\vee\vee}$.
I tried to prove this using the following way:
Given a nonzero element $v \in V$, consider the points that $\lambda(v)$ doesn't vanish, denoting as $X_{v}$. Then by $V$ being base-point-free, $X_v$ covers $X$ as $v$ running over all nonzero element in $V$. And $X_v$ is a trivilization of $\mathscr{L}$. Now define
$$ X_v \to \operatorname{Spec} \left( (\operatorname{Sym}V^{\vee\vee})_{v^{\vee\vee} } \right)_0 \hookrightarrow \Bbb{P}_n^{\vee\vee} $$
by
$$ \left( (\operatorname{Sym}V^{\vee\vee})_{v^{\vee\vee} } \right)_0 \to \Gamma(X_{v}, \mathscr{O}_X) $$
taking $w^{\vee\vee}/v^{\vee\vee}$ to $\lambda(w)/\lambda(v)$. But this seems only working when $V$ is finite dimensional, in which case $V$ is isomorphism to $V^{\vee\vee}$ and every element in $\operatorname{Sym}V^{\vee\vee}$ has the form $w^{\vee\vee}$.
Does the result here hold for general case (since Vakil put the condition "finite-dimensional" inside a parentheses)? Although only working in finite dimensional case but I am still kind of interesting in it. And if so, from the argument for the infinite case maybe I can find out if there any problem in my argument above.
Thank you in advance!