Are Duality and the Inner-Product inextricably linked? In linear algebra, the dual $A^*:W^*\to V^*$ of a map $A:V\to W$ is defined by
$$\langle Av,w\rangle=\langle v,A^*w\rangle$$
Does this mean that something like $A^*$ has no meaning without the context of an inner-product space?

If $A:V\to W$ is linear, there is a map from $W^*$ to $V^*$ that sends $f$ in $W^*$ to the composition $f\circ A$ in $V^*$. (So if $v\in V$, then $f\circ A$ maps $v$ to $f(Av)$.) 
This map from $W^*$ to $V^*$ is linear, and is denoted by $A^*$. It is defined without reference to any inner product or pairing.
