Remark: A “projective variety” means an irreducible algebraic set of $\mathbb{P}^n(k)$, where k is an algebraic closed field throughout the question.
Recently, I've run into to some problems in reading Fulton's Algebraic Curves. In Section 4.2, he constructs the following definitions:
Let V be a projective variety, then:
$The\ homogeneous\ coordinate\ ring\ of\ V\ :\ \Gamma_{h}(V) = k[X_{1},...,X_{n+1}]/I(V)$
$The\ homogeneous\ function\ field\ of\ V\ :\ k_{h}(V) = the\ quotient\ field\ of\ \Gamma_{h}(V)$
$The\ function\ field\ of\ V\ :\ k(V) = \{z \in k_{h}(V) \mid for\ some\ forms\ f,g \in \Gamma_{h}(V)\ of\ the\ same\ degree, z = f/g\}$
It is easy to see that every element of $k(V)$ can be regarded as a function of $V$. But what I really want to ask is whether $k(V)$ collect "all functions of $V$"?
To make it clear, let me define what so-called "all functions of $V$" is first:
$F(V) = All\ functions\ of\ V \ :=\ \{z \in k_{h}(V) \mid If\ there\ exist\ f \in \Gamma_{h}(V),\ g \in \Gamma_{h}(V)\ and\ (x_{1},...,x_{n+1}) \in \mathbb{P}^n\ such\ that\ z = f/g\ and\ g(x_{1},...,x_{n+1}) \ne 0,\ then\ there\ exist\ f_{\lambda}\ \in \Gamma_{h}(V),\ g_{\lambda} \in \Gamma_{h}(V)\ such\ that\ z = f_{\lambda}/g_{\lambda},\ g_{\lambda}(\lambda x_{1},...,\lambda x_{n+1}) \ne 0\ and\ f(x_{1},...,x_{n+1})/g(x_{1},...,x_{n+1}) = f_\lambda(\lambda x_{1},...,\lambda x_{n+1})/g_\lambda(\lambda x_{1},...,\lambda x_{n+1})\ for\ all\ \lambda \in k \}$
I think $F(V)$ is the precise description about "all function of $V$". And I tried to prove that $F(V) = k(V)$ or $F(V) \ne k(V)$, but I can't. So I wonder if we can make clear the relation between $F(V)$ and $k(V)$ or if I made some mistakes in the notion about "all functions of $V$".