Geometric visualization of covector? How could I geometrically visualize a linear functional?
 A: I think the nicest visualization is the set of hyperplanes where it attains integer values (yes, technically one hyperplane, like that one where its value is $1$, suffices to uniquely specify it, but the whole set is IMHO more intuitive, and that's the whole point of visualization, isn't it?). They are the equal value hypersurfaces of the linear function. Of course you'll also have to specially mark which direction is "up" (e.g. by giving the positive value surfaces a different color). The value when applying to a vector is the number of the hyperplane where it ends (if it ends in between, you simply interpolate, or alternatively think of a finer-grained set (like the millimeter lines in between centimeter lines on a ruler). Or alternatively, the value is the number of planes it intersects (again, with interpolation/finer grained subsets).
You immediately see the linearity from the fact that they are equal spaced and parallel; a twice as long vector goes through twice as many hyperplanes. Also, multiplication with a scalar is quite obvious: If you multiply it with $m/n$, you get $m$ planes in the same space you've gotten $n$ planes before. And addition of parallel covectors is basically stuffing the planes from both covectors in the same space, but equally spaced.
Addition of non-parallel covectors is only slightly more involved: You basically have to draw hyperplanes through all the intersections of the hyperplanes of the two added covectors. However you've got to be careful to do it correctly: The zero plane of the sum covector goes through the crossings of positive hyperplanes from one, and negative hyperplanes from the other term.
Note that for this representation you do not need to assume an inner product (if you do so on a space where there's not a natural choice of inner product, your intuition might be misled by giving significance to absolute lengths and angles in your visualization, which are just meaningless in a space without inner product).
A: In the finite dimensional case you always have that a linear functional corresponds to a unique vector through the inner product $\psi(x) = \langle x,y \rangle$ for some $y$, this passes to the infinite dimensional case when you have an inner product space (Riesz Representation Theorem)
A: I give you an example of geometric visualization: consider, for example, any vector $v=(v_1,v_2)\in\mathbb R^2$ and the linear functionals
$$\varphi_1: \mathbb R^2\rightarrow \mathbb R$$
$$\varphi_2: \mathbb R^2\rightarrow \mathbb R$$
given by $\varphi_1(v_1,v_2)=v_1$ and $\varphi_2(v_1,v_2)=v_2$. You can visualize them as the functionals which provide you with the projections of the given vector $v$ to the $x$-axis, resp. the $y$-axis.
A: In ${\Bbb{R}}^2$, being $f:{\Bbb{R}}^2\to {\Bbb{R}}$ linear, its level curves are lines perperdicular to the vector given by ${\rm graf} f$.
In ${\Bbb{R}}^3$, being $f:{\Bbb{R}}^3\to {\Bbb{R}}$ linear, its level surfaces are planes perperdicular to the vector given by ${\rm graf} f$.
More details:
I said that the level sets are perpendicular to the gradient is because in the case of the domain is 
${\Bbb{R}}^n$ 
then your covector must be of the form
$$f(x_1,x_2,...,x_n)=a_1x_1+a_2x_2+\cdots+a_nx_n,$$
for some constants $a_i$ in $\Bbb{R}$.
Then the gradient also is  going to be  constant
$${\rm grad}f=[a_1,a_2,...,a_n].$$
At a level set $f^{-1}(c):=\{q=(x_1,...,x_n):f(q)=c\}$ 
consider a point $p\in f^{-1}(c)$ and a curve $\alpha(t)$ such that
$$\alpha(0)=p$$
and
$$\alpha'(0)=V$$
It is known that $V$ is tangent to the curve $\alpha$ at $p$.
So by differentating
$f\circ\alpha(t)=c$ you are going to get (after evaluating at $t=0$)
$${\rm grad}f(\alpha(0))\cdot\alpha'(0)=0$$
i.e.
$${\rm grad}f(p)\cdot V=0$$
That is $V$ is perpendicular to $[a_1,...,a_n]$.
So, the curve $\alpha$ being in $f^{-1}(c)$ has a tangent $\alpha'$ always perpendicular to $[a_1,...a_n]$
so the curve $\alpha$ is a line in $f^{-1}(c)$.
Being this for each $p$ and each $V$ then  the level set $f^{-1}(c)$  is a hyperplane.
With $n=2$ a level line. With $n=3$ a level plane,... et cetera.
A: Here's a picture of the scalar field pulled back onto the plane it's assigning values to.

So what are the basis elements of the covector? The assignment here is $$1 \cdot \mathrm{vert} + 4 \cdot \mathrm{horiz}$$; so I chose basis elements that match the $\vec{x}$'s basis elements. If you changed the 4 and 1 you'd shift the way these level sets "slant"...... (and this would be isomorphic to choosing the same rates-of-increase but changing the direction of rate-of-increase (different basis).

You can also think of moving the covector as changing the weightings of a weighted average (sum, really, but weighted averages are more common in everyday talk).
A: A non-zero linear functional can be visualized a hyperplane (a line in $\mathbb{R}^2$, a plane in $\mathbb{R}^3$). The duality of vectors and covectors can be visualized using duality of points and hyperplanes in projective geometry.
Notation.
We will use a row vector $\pmb{v} = 
\begin{pmatrix}
v^1  \\
\vdots \\
v^n \\
\end{pmatrix} \in \mathbb{R}^n$ to represent coordinates of a point and a column vector $\pmb{w}=
\begin{pmatrix}
w_1,  \cdots,  w_n \\
\end{pmatrix} \in \mathbb{R}_n$ to represent hyperplane. We define a usual product of a row and column as $\pmb{w}\cdot\pmb{v}= \sum_{i=1}^n w_i v^i$.
A non-zero $\pmb{w} \in \mathbb{R}_n$ determines a hyperplane
$\pmb{w}_H= \{ \pmb{v} \in \mathbb{R}^n : \pmb{w}\cdot\pmb{v} = 0 \}$, while a non-zero $\pmb{v} \in \mathbb{R}^n$ determines a line $\pmb{v}_P = \{u\pmb{v} \in \mathbb{R}^n: u\in \mathbb{R}\}$ (a point in projective space), the projective space is $\mathbf{P}(\mathbb{R}^n)=\{\pmb{v}_P: \pmb{v}\in \mathbb{R}^n, \pmb{v}\neq 0\}$. We will write $\pmb{v}_*=\pmb{v}$ to stress that $\pmb{v}$ is interpreted as a point in $\mathbb{R}^n$.
Vectors visualized in projective space.
Let's first interpret a vector $\pmb{v}$ in $\mathbb{R}^n$ in a projective space $\mathbf{P}(\mathbb{R}^n\times \mathbb{R})$. In general, any vector space $V$ is isomorphic to $\operatorname{Hom}(\mathbb{R}, V)$. So instead of $\pmb{v} \in \mathbb{R}^n$ we can take a linear map $\pmb{v}:\mathbb{R} \to \mathbb{R}^n, u\mapsto u\pmb{v}$ and note that the set
$\left\{\left(\begin{matrix}u\pmb{v} \\ u\end{matrix}\right)\in\mathbb{R}^n\times\mathbb{R}: u\in\mathbb{R}\right\}$, which is essentially a graph of $\pmb{v}$, is a point in $\mathbf{P}(\mathbb{R}^n\times \mathbb{R})$.
It is is more convenient to think about $\mathbf{P}(\mathbb{R}^n \times \mathbb{R})$ in terms of a decomposition into to two copies of $\mathbb{R}^n$, the first copy represents usual points in $\mathbb{R}^n$ and the second copy represents points at infinity. More precisely, a point in $\mathbf{P}(\mathbb{R}^n \times \mathbb{R})$ is either $\begin{pmatrix}\pmb{v} \\ 1\end{pmatrix}_P$ or $\begin{pmatrix}\pmb{v}' \\ 0\end{pmatrix}_P$ for some $\pmb{v}, \pmb{v}'\in \mathbb{R}^n, \pmb{v'} \neq \pmb{0}$ and it can be identified with a point $\pmb{v}_* =\pmb{v} \in \mathbb{R}^n$ or a projective point $\pmb{v'}_P \in \mathbf{P}(\mathbb{R}^n)$. In other words, $\mathbb{R}^n \cup \mathbf{P}(\mathbb{R}^n)$ can be identified with $\mathbf{P}(\mathbb{R}^n \times \mathbb{R})$. Then a vector $\pmb{v}$ can be geometrically visualized as having an origin $\pmb{0}_*$ and an end $\pmb{v}_*$ and a direction $\pmb{v}_P$. The point $\pmb{v}_*= \pmb{v}(1)$ is the essential part.
Covectors visualized in projective space.
Let's interpret a non-zero linear functional (covector) given by a $\pmb{w}\in \mathbb{R}_n$, $\pmb{w}:\mathbb{R}^n \to \mathbb{R}, \pmb{v}\mapsto\pmb{w}\cdot \pmb{v}$ in the projective space $\mathbf{P}(\mathbb{R}^n \times \mathbb{R})$. The graph of this linear functional $\left\{\begin{pmatrix}
\pmb{v}  \\
\pmb{w}\cdot \pmb{v} \\
\end{pmatrix}:\pmb{v}\in\mathbb{R}^n\right\}$ is a hyperplane in $\mathbf{P}(\mathbb{R}^n \times \mathbb{R})$. The coordinates of this hyperplane are given by $(\pmb{w},-1)$ as $(\pmb{w},-1)\cdot \begin{pmatrix}
\pmb{v}  \\
\pmb{w}\cdot \pmb{v} \\
\end{pmatrix} = \pmb{w}\cdot\pmb{v} - 1\cdot\pmb{w}\cdot\pmb{v} = 0$. But in the decomposition   $\mathbb{R}^n \cup \mathbf{P}(\mathbb{R}^n)$ this defines a hyperplane in $\mathbb{R}^n$: $\pmb{w}^*=\{\pmb{v}\in\mathbb{R}^n: \pmb{w}\cdot\pmb{v}=1\}$. Likewise the graph of the zero linear functional $\pmb{0}$ is a hyperplane in $\mathbf{P}(\mathbb{R}^n \times \mathbb{R})$, which in turn we identify with $\mathbf{P}(\mathbb{R}^n)$, a hyperplane at infinity. A covector $\pmb{w}$ can be geometrically visulized as having an origin $\pmb{0}^*=\mathbf{P}(\mathbb{R}^n)$, an end $\pmb{w}^*$ and a direction $\pmb{w}_H$. Again $\pmb{w}^*=\pmb{w}^{-1}(1)$ is the essence of this interpretation.
To sum up:
$$\begin{array}{c|c} 
 \text{vector} & \text{covector} \\ \hline
\text{a linear map: } \pmb{v}:\mathbb{R} \to \mathbb{R}^n & \text{a linear map: } \pmb{w}: \mathbb{R}^n \to \mathbb{R} \\ \hline
\text{determined by } \pmb{v}(1) & \text{determined by } \pmb{w}^{-1}(1) \\ \hline
\text{a column } \pmb{v} \in \mathbb{R}^n &  \text{a row } \pmb{w} \in \mathbb{R}_n \\ \hline
\text{a point } \pmb{v}_* &  \text{a hyperplane } \pmb{w}^*  \\ \hline
\text{its origin is } \pmb{0}_* = \pmb{0}&  \text{its origin is } \pmb{0}^*=\mathbf{P}(\mathbb{R}^n)  \\ \hline
\text{its direction is } \pmb{v}_P &  \text{its direction is } \pmb{w}_H  \\ \hline
\text{upper index (up and down)} & \text{lower index (left to right)} \\ \hline
\end{array}$$
The addition of vectors and covectors can be explained with the  Parallelogram Law and its dual version. For simplicity, I will illustrate this in $\mathbb{R}^2$. I've made an interactive geometric visualization available in desmos of addition and scalar multiplication of vectors and covectors.
Geometric addition of vectors.
Let $\pmb{v, w} \in \mathbb{R}^2$ be two independent vectors. Their directions are $\pmb{v}_P$ and $\pmb{w}_P$. We find a line $\pmb{v}_*\pmb{w}_P$ that passes through $\pmb{w}_P$ and $\pmb{v}_*$ (a line parallel to $\pmb{0}_*\pmb{w}_*$) and another line $\pmb{w}_*\pmb{v}_P$ that passes through $\pmb{v}_P$ and $\pmb{w}_*$ (a line parallel to $\pmb{0}_*\pmb{v}_*$). The point where these two lines intersect represents the sum of $\pmb{v}_*$, $\pmb{w}_*$, it is $(\pmb{v} + \pmb{w})_*$. In the case of $\mathbb{R}^2$ the lines are given by these covectors:
$$\pmb{v}_*\pmb{w}_P = \left(\frac{w^2}{\det(\pmb{v},\pmb{w})},\frac{-w^1}{\det(\pmb{v},\pmb{w})}\right)^*$$
$$\pmb{w}_*\pmb{v}_P =\left(\frac{-v^2}{\det(\pmb{v},\pmb{w})},\frac{v^1}{\det(\pmb{v},\pmb{w})}\right)^*$$

Geometric addition of covectors.
Let $\pmb{v, w} \in \mathbb{R}_2$ be two independent covectors. Their directions are respectively $\pmb{v}_H$ and $\pmb{w}_H$.  We find the point where $\pmb{v}^*$ and $\pmb{w}_H$ intersect ($\pmb{v}^*\cap \pmb{w}_H$) and another point where $\pmb{w}^*$ and $\pmb{v}_H$ intersect ($\pmb{w}^*\cap \pmb{v}_H$). The line passing through these two points represents a sum of $\pmb{v}^*$ and $\pmb{w}^*$, it is $(\pmb{v} + \pmb{w})^*$. In the case of $\mathbb{R}^2$, these points are given by these vectors:
$$\pmb{v}^*\cap \pmb{w}_H = \frac{1}{\det(\pmb{v},\pmb{w})}\begin{pmatrix}w_2 \\ -w_1\end{pmatrix}_*$$
$$\pmb{w}^*\cap \pmb{v}_H = \frac{1}{\det(\pmb{v},\pmb{w})}\begin{pmatrix}-v_2 \\ v_1\end{pmatrix}_*$$

