Can the Dual space be thought of as an extension of $\mathbb{R}^l$ As $\mathbb{R}$ is an extension of $\mathbb{Q}$, can the Dual space be thought of as an extension of $\mathbb{R}^l$?  
Where the dual space of $\mathbb{R}^l$ is the set of all continuous linear functions
I am having trouble imagining the dual space as it is the first function space I have encountered.  
Could we write the set notation of the dual space as something like the set of ordered pairs that are linear functions:
$$ \{ \{x_0, f(x_0), \{x_1,f(x_1)\}, \dots, \{y_0, g(y_0)\}, \dots \; | \; functions \; are \; linear\} ?$$
If I may also ask, is there a simpler function space to imagine as one would a Metric space or Vector space?  Apologies for the multiple questions, clarification on any of these would be helpful, thank you!
 A: Actually the dual space $(\mathbb R^l)^*$ of $\mathbb R^l$ looks very similar to $\mathbb R^l$. Explicitly, it's common to represent the elements of $\mathbb R^l$ as $l$-dimensional column vectors. Given an $l$-dimensional row vector $r$, you can define a linear map $\mathbb R^l\rightarrow\mathbb R$ by $v\mapsto rv$. In fact $v$ is determined by this linear map, and conversely every linear map can be produced this way. So you can think of $(\mathbb R^l)^*$ as the space of $l$-dimensional row vectors.
This is a good way to think about $(\mathbb R^l)^*$ if you want to build some intuition about what the space looks like. We can switch between row and column vectors by transposing, but it's useful to distinguish between them because some operations make sense for row vectors but not column vectors, and vice-versa. Similarly there are isomorphisms between $\mathbb R^l$ and $(\mathbb R^l)^*$, but it's useful to think of them as distinct spaces.
The above doesn't help if you are more concerned with treating a set of functions as a vector space. Maybe as a simpler example you could consider the set of all functions from some set $X$ to $\mathbb R$ (with no conditions). You could define addition and scalar multiplication operations and check that the various vector space axioms hold.
In fact $(\mathbb R^l)^*$ is a linear subspace of the space of all functions from $\mathbb R^l$ to $\mathbb R$. However the latter is infinite dimensional, so this may not be the easiest way to understand $(\mathbb R^l)^*$.
