The concept of dual spaces is used frequently in abstact mathematics, but also has some practical applications. Consider a 2D vector space ${\displaystyle V=R^{2}}$ on which a differentiable function ${\displaystyle f}$ is defined. As an example, ${\displaystyle V}$ can be the Cartesian coordinates of points in a topographic map and ${\displaystyle f=f(c_{1},c_{2})}$ can be the ground altitude which varies with the coordinate ${\displaystyle x=(c_{1},c_{2})}$ . According to theory, the infinitesimal change ${\displaystyle df}$ of ${\displaystyle f}$ at the point ${\displaystyle x=(c_{1},c_{2})}$ as a consequenece of changing the position an infintesimal amount ${\displaystyle dx=(dc_{1},dc_{2})\in V}$ is given by

${\displaystyle df=dc_{1}{\frac {df(x)}{dc_{1}}}+dc_{2}{\frac {df(x)}{dc_{2}}}}$

the scalar product between the vector ${\displaystyle dx}$ and the gradient of ${\displaystyle f}$. Clearly, ${\displaystyle df}$ is a scalar and since it is constructed as a linear mapping on ${\displaystyle dx}$, by computing its scalar product with ${\displaystyle \nabla f(x)}$, it follows from the above defintion that ${\displaystyle \nabla f(x)}$ is an element of ${\displaystyle V^{\star }}$.

From the outset, both vectors ${\displaystyle dx}$ and ${\displaystyle \nabla f(x)}$ can be seen as elements of ${\displaystyle R^{3}}$. Why is a dual space needed? What is the difference between ${\displaystyle V}$ and ${\displaystyle V^{\star }}$ in this case?

To see the difference between ${\displaystyle V}$ and ${\displaystyle V^{\star }}$, remember that in practice both vectors ${\displaystyle dx}$ and ${\displaystyle \nabla f(x)}$ must be expressed as a set of three real number which are their coordinates relative to some basis of ${\displaystyle R^{3}}$. Intuitively we may choose to use an orthogonal basis, with normalized basis vectors which are mutually perpendicular. Let ${\displaystyle E=\{e_{1},e_{2},e_{3}\}}$ be a such a basis for ${\displaystyle R^{3}}$. This means that ${\displaystyle dx}$ can be written as

${\displaystyle dx=dx_{1}e_{1}+dx_{2}e_{2}+dx_{3}e_{3}}$

where ${\displaystyle dx_{1},dx_{2},dx_{3}}$ are the (infinitesimal) coordinates of ${\displaystyle dx}$ in the basis ${\displaystyle E}$. Similiarly, ${\displaystyle \nabla f(x)}$ can be written as

${\displaystyle \nabla f(x)=\nabla _{1}f(x)e_{1}+\nabla _{2}f(x)e_{2}+\nabla _{3}f(x)e_{3}}$

where ${\displaystyle \nabla _{1}f(x),\nabla _{2}f(x),\nabla _{3}f(x)}$ are the coordinates of ${\displaystyle \nabla f(x)}$ in the basis ${\displaystyle E}$. Given that the coordinates of both