> the best analogy is a basis change in a vector space: It's closer than an analogy. FT is essentially a map of basis into a dual space, but this dual is non u

> the best analogy is a basis change in a vector space: It's closer than an analogy. FT is essentially a map of basis into a dual space, but this dual is non uniquely isomorphic with the original so given an arbitrary choice of isomorphism you can turn it into "just" a change of basis.