@2015-8-30

Basic

It pays to study functions. In fact, functions appear at times to be more basic than their domains. In the context of linear algebra, the natural class of functions is linear transformations, or linear maps from one vector space to another. Among all real vector spaces, there is one that seems simplest, namely the one-dimensional vector space of the real numbers R. This leads us to examine a special type of linear transformation on a vector space, those that map the vector space to the real numbers, the set of which we will call the dual space. Dual spaces regularly show up in mathematics.

Let V be a vector space. The dual vector space, or dual space, is:

$$V^*=\{\text{ linear maps from } V \text{ to the real numbers } R\} =\left\{v^*:V \to R\left|v^* \text{ is linear}\right.\right\}.$$

You can check that the dual space $V^*$ is itself a vector space.

Let $T: V\to W$ be a linear transformation. Then we can define a natural linear transformation

$$T^*: W^*\to V^*$$

from the dual of W to the dual of V as follows. Let $w^*\in W^*$. Then given any vector w in the vector space W, we know that $w^* (w)$ will be a real number. We need to define $T^*$ so that $T^* \left(w^*\right)\in V^*$. Thus given any vector vV, we need $T^* \left(w^*\right) (v)$ to be a real number. Simply define

$$T^*\left(w^*\right)(v) = w^*(T(v)).$$

By the way, note that the direction of the linear transformation $T:V \to W$ is indeed reversed to $T^* : W^* \to V^*$. Also by "natural", we do not mean that the map $T^*$ is "obvious" but instead that it can be uniquely associated to the original linear transformation T.

Such a dual map shows up in many different contexts. For example, if X and Y are topological spaces with a continuous map $F:X\to Y$ and if $C(X)$ and $C(Y)$ denote the sets of continuous real-valued functions on X and Y, then here the dual map

$$F^*:C(Y)\to C(X)$$

is defined by $F^*(g)(x) = g(F(x))$, where g is a continuous map on Y.

Attempts to abstractly characterize all such dual maps were a major theme of mid-twentieth century mathematics and can be viewed as one of the beginnings of category theory.