Dot product between two vectors or vector and 1-form? When we take a dot product between two vectors of a vector space, we actually "act" by a 1-form (dual vector) on a vector. So why most books define the dot product between vectors? Of course with the help of a metric we can take a dot product between two vectors, but technically the metric converts one of the two vectors to a 1-form.
 A: An inner product is definitely a certain bilinear map
$$\langle \cdot, \cdot \rangle: V \times V \longrightarrow \Bbb R$$
which takes two vectors as its arguments. If you're thinking in terms of a metric tensor $g$, then $g$ is a type $(0,2)$-tensor, which means it has two vector arguments (no covector arguments).
What you're thinking of is the fact that a choice of inner product $\langle \cdot, \cdot \rangle$ on a vector space $V$ gives an isomorphism
$$\varphi: V \longrightarrow V^\ast,$$
$$v \mapsto \langle v, \cdot \rangle.$$
In this sense we can identify the inner product as the action of a covector on a vector:
$$\langle u, v \rangle = \varphi(u)(v).$$
In terms of components of the metric tensor, this is equivalent to
$$\langle u, v \rangle = g_{ij} u^i v^j = u_j v^j.$$
An element of $V^\ast$ acts on $V$ in the same way no matter what basis we choose, but there is no canonical identification of $V$ with $V^\ast$. Without a metric, the lowered index components $u_i$ of a vector $u$ make no sense.
