Dual Space of $\mathbb{R}^n$

Question

I am trying to understand dual spaces. Suppose I have the Euclidean vector space $\mathbb{R}^n$ over the field $\mathbb{R}$. Elements of this space are column vectors $\boldsymbol{x}\in\mathbb{R}^{n \times 1}$. A possible linear functional can be defined via matrix multiplication, given a row vector $\boldsymbol{a}\in\mathbb{R}^{1\times n}$ $$ f_{\boldsymbol{a}}: \mathbb{R}^n \to \mathbb{R} \qquad \text{s.t.} \qquad \boldsymbol{x} \mapsto f_{\boldsymbol{a}}(\boldsymbol{x}) = a_1x_1 + \cdots a_nx_n $$ This means that a possible dual space of $\mathbb{R}^{n}$ is the space of all such mappings $$ (\mathbb{R}^n)^* = \left\{f_{\boldsymbol{a}}: \mathbb{R}^n \to \mathbb{R} \, \mid \, \boldsymbol{a}\in \mathbb{R}^{1\times n}\right\} $$ However, on Wikipedia they say "its dual space is typically written as the space of rows of $n$ real numbers". How come this difference? Is it becomes there is a unique correspondence between each vector $\boldsymbol{a}\in\mathbb{R}^{1 \times n}$ and each function $f_{\boldsymbol{a}}$? In that case, how does one formalize this reasoning? The only thing that bugs me is that the elements of the dual space must be linear functionals. While $f_{\boldsymbol{a}}$ are definitely linear funcitonals, I'm pretty sure $\boldsymbol{a}$ are not.

Answer 1

This means that a possible dual space of Rn is the space of all such mappings

This statement is a little off. There is not "a possible" dual space. There is only one dual space. Since the set you write there contains every possible linear functional from $\mathbb{R}^n$ to $\mathbb{R}$ it is the dual space of $\mathbb{R}^n$.

And you're right the dual space doesn't consist of row vectors because, as you said, a row vector is not itself a function. But as you point out there is a one-to-one correspondence between row vectors a linear functions from $\mathbb{R}^n$ to $\mathbb{R}$. So when we want to refer to a function $f$ we always do it by specifying vector $\mathbf{a}$ such that $f = f_{\mathbf{a}}$.

So the wikipedia article wasn't trying to say the dual space actually consists of row vectors. It's just saying this is how we typically represent the functions in the dual space.