User:KYN/WhyDualSpace

Why do we need dual spaces?

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

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

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

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

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

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

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

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

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