In undergraduate linear algebra, the concept of a dot product, generalized to the inner product on an inner product space, is introduced fairly early as a way to multiply 2 vectors together to get a scalar.
As one continues through an undergraduate curriculum to physics, however, this notion gets largely replaced by that of a dual space, and the inner product of vectors becomes replaced by the notion of multiplying vectors with a member of their dual space.
This seems to me to be two realizations of the same motivation: multiplying 2 vectors together to get a scalar. I understand that there are formal mathematical differences between the two, and the Riesz representation theorem gives a map between them, but I'm struggling with why we need both to exist to begin with.
To help me understand the fundamental conceptual differences between the two, in physics, why does "multiplying 2 vectors together" take the form of the inner product in some circumstances, and dual spaces in others?
E.g., I've seen bra/ket multiplication represented both by an inner product and dual space formalism, and it seems the structures are somewhat redundant if they have the same utility.
General relativity seems to prefer using dual spaces, but it's not clear to me why the extra structure is necessary and, for example, why we couldn't inner product vectors at a point with other members of the same vector space to achieve the same physical results rather than formalizing the physics in a dual space construct.
I'm sure there are more examples, where physics needs a concept of "multiplying 2 vectors together" and the community decides to say the second operand lives in its own dual vector space, rather than having it "share" an inner product space with the first operand. Why is this natural? In what cases is one representation of this idea more "powerful" or "natural" than the other?