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