How to identify "naturally" two subspaces? I am self-studying  Hoffman and Kunze's book Linear Algebra. This is exercise $1$ from page $111$.
If $W$ is a subset of a vector space $V$, we define $W^{0}=\{f\in V^{\star}|f(w)=0 \operatorname{for all}w\in W\}.$

Let $n$ be a positive integer and $\mathbb{F}$ be a field. Let $W$ be
  a set of all vectors $(x_{1},\ldots,x_{n})$ in $\mathbb{F}^{n}$ such
  that $x_{1}+\cdots+x_{n}=0$.
a) Prove that $W^{0}$ consists of all linear functionals $f$ of the
  form $$f(x_{1},\ldots,x_{n})=c(\sum_{j=1}^{n}x_{j}).$$
b) Show that the dual space $W^{\star}$ of $W$ can be "naturally"
  identified with the linear functionals
  $$f(x_{1},\ldots,x_{n})=c_{1}x_{1}+\cdots+c_{n}x_{n}$$ on
  $\mathbb{F}^{n}$ which satisfy $c_{1}+\cdots+c_{n}=0$.

My approach for part a) First, we note that $e_{i}-e_{j}\in W$ for all $i,j\in\{1,2,\ldots,n\}$. If $f\in W^{0},$ then we have $f(e_{i})-f(e_{j})=f(e_{i}-e_{j})=0.$ Therefore $f(e_{i})=f(e_{j})$ and we have $f(e_{i})=c$, for all $i\in \{1,2,\ldots,n\}$, where $c$ is a constant. Therefore, if $w=(x_{1},\ldots,x_{n})\in W$, then $f(x_{1},\ldots,x_{n})=c(\sum_{j=1}^{n}x_{j}).$ The converse is easy. 
I was not able to solve the part b).
 A: Let
\[
H = \{f \in (\mathbb F^n)^* \mid f(x_1, \ldots, x_n) = \sum c_ix_i \text{ and } \sum c_i = 0 \}.
\]
There's a map $(\mathbb F^n)^* \to W^*$ given by restriction—if you like, this is the dual $i^*$ of the inclusion $i\colon W \to \mathbb F^n$. Restrict this to a map $H \to W^*$ and show that you obtain an isomorphism. For this, you could use the fact that the kernel of $i^*$ is precisely $W^0$. What is $W^0 \cap H$?
I take the word "natural" here to mean, "Don't just identify $W$ and $H$ because they have the same dimension." The word often means "basis-free" (or functorial), but here it seems to me that we've chosen a basis by merely writing $\mathbb F^n$. In general, if I have a vector space $V$ and a subspace $W \subset V$ then the dual of $W$ is most naturally identified with the quotient $V^*/W^0$. (In the same spirit, can you find something naturally isomorphic to $(V/W)^*$?)
A: This does not address OP's concern, but this does something irrelevant and trivial. 

The part $(b)$ is largely about writing the details down:
$$W=\{\underline w =(w_1,\dots,w_n) \mid \sum_{i=1}^nw_i=0\} \tag{1}$$ $$W^{\ast}=\{f \mid f\;\; \mbox{linear function from } W \;\;\mbox{to}\;\;\Bbb F\} \tag{2
}$$
Let's also define $f_{\underline x}:W \to \Bbb F$ given by $f_{\underline x}(\underline w)=\underline x \cdot \underline w$. Let's use this to define the following map:
$$\begin{align}W &\to W^\ast \\ \underline x&\mapsto f_{\underline x}\end{align}$$
Proving that the above map is an injective homomorphism identifies $W$ with $W^\ast$. Since, $\dim W=\dim W^\ast$, we also have that this map is an isomorphism. 
But OP wants an altogether different isomorphism, so, I GOOFED IT UP. 
