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?