For a graph, is its degree sequence a function defined on its vertex set? In West's Introduction to Graph Theory

1.3. Vertex Degrees and Counting
1.3 1. Definition. The degree of vertex $v$ in a graph $G$, written $d_G(v)$ or $d(v)$, IS the number of edges incident to $v$, except
that each loop at $v$ counts twice.
GRAPHIC SEQUENCES
Next we consider all the vertex degrees together.
1.3.27. Definition. The degree sequence of a graph is the list of vertex de­grees .
Every graph has a degree sequence, but which sequences occur? That is,
given nonnegative integers $d_l , . . . , d_n$, is there a graph with
these as the vertex degrees? The degree-sum formula implies that
$\sum_i d_i$ must be even.  When we allow loops and multiple edges,
the obvious necessary condition is also sufficient:
1.3.28. Proposition. The nonnegative integers $d_l , . . . , d_n$ are the vertex de­grees of some graph if and only if $\sum_i d_i$ is even.

In 1.3.1 definition, is $d_G$ a function defined on $V(G)$?
Are $d_G$ and $d_l , . . . , d_n$  in 1.3.27 definition  equivalent to each other?
Why do we need two representations of the same thing? (Why do we need the concept of degree sequence of a graph, instead of just its $d_G$?)
Thanks.
 A: No, the degree sequence is not a function on $V(G)$, although it involves a function on $V(G)$ indirectly in its definition, namely the function $d : V(G) \to \{0,1,2,3,...\}$ which assigns to each $v \in V(G)$ its degree $d(v)$.
Amongst all enumerations of the vertex set $V_G$, choose an enumeration $v_1,v_2,...,v_n$ of $V_G$ having the property that
$$d(v_1) \ge d(v_2) \ge d(v_3) \ge ... \ge d(v_n)
$$
Then the degree sequence $d_1,d_2,d_3,...,d_n$ is defined by $d_i = d(v_i)$. So in the degree sequence, you have recorded the degrees in order, but you have lost the information of what degree goes with what vertex. So no, the function $d$ and the degree sequence $d_1,d_2,...,d_n$ are not equivalent to each other.
To address your question of why we need two representations of the same thing, well, because sometimes we want to ask a question or formulate a statement about the degrees themselves, such as Proposition 1.3.28, so that's what the degree sequence is used for. Other times we want to know the actual degrees of each vertex, that's what the degree function $d(v)$ is for.
A: What you’ve written $d_l$ should be $d_1$. The numbers $d_1,\ldots,d_n$ are simply non-negative integers; they may or may not be the degrees of the vertices of some graph, and even if they are, the sequence, unlike the function $d_G$, does not actually associate each of them with a vertex. The question is which such sequences actually are the values of the function $d_G$ on the vertex set $V(G)$ of some graph $G$.
A: 
In 1.3.1 definition, is $d_G$ a function defined on $V(G)$?

Yes.

Are $d_G$ and $d_l,\ldots,d_n$ in 1.3.27 definition equivalent to each other?

No. The degree sequence forgets which value corresponds to which vertex. So it's not a function anymore, because it's not a relation between inputs and outputs — it's only a list of numbers. We should think of it as a new object, a finite nondecreasing sequence of nonnegative integers, built from the given graph as a whole.

Why do we need two representations of the same thing? (Why do we need the concept of degree sequence of a graph, instead of just its $d_G$?)

For example, in some applications where we'd want to model things with graphs, we may come up with a desired degree sequence first, and then build a graph from it.
