What is the relation between dual spaces and inner product?

What is the relation (if any) between dual spaces and inner product? As far as I understand the dual space of a vector space is the set of all linear mappings from the vector set to the field over which the space is defined. But the definition of the inner product is a bilinear mapping of two vectors to a scalar. It sounds to me like if we had defined the same thing twice, in two different ways, is that so?

If the answer is yes, and given that every space has a dual space, does that mean that every vector space is automatically an inner product space? Moreover, if the polarization identity can be used to define a norm from an inner product, are all vector spaces inner normed spaces?

I am sure I'm misunderstanding some definition, but I'm totally lost here. Any help?

Not every vector space is an inner product space because not every norm satisfies the parallelogram law. As classic counterexamples, consider the the spaces $\mathbb{R}^n$ under the $1$-norm (aka taxicab norm) and the $\infty$-norm (aka maximum/supremum norm).

If you are given an inner-product space (aka Hilbert space), then there is indeed a strong connection between the dual space and the inner product. This result is known as the Riesz representation theorem. Note, however, that his does not mean that we've "defined the same thing twice". The dual space is the set of all linear mappings to the scalar field, whereas the inner product of an inner product space is a particular (bilinear) map on two vectors to the scalar field.

If you have an inner product, then you have an isomorphism from $V$ to its dual $V^*$ given by $v\mapsto \langle v,-\rangle$. But, there are vector spaces that are not inner product spaces. For that see here.