Recently I've been self-studying tensor analysis through the book titled "An Introduction to Tensors and Group Theory for Physicists," by Nadir Jeevanjee. I found it to be very intuitive, up until the part about Dual Vectors and Dual Spaces.
I understand these are really important concepts, like in Quantum Mechanics, for example, but it isn't making too much sense to me. From what I understand, a dual vector is just a linear functional; it eats up a vector and spits out a number. Am I wrong?
How is a linear functional different than any other function that I've seen in a class like Calculus 3? And what exactly does any of this have to do with tensors?
I'm not looking for an answer that's terribly mathematically rigorous, but just intuitive, or an answer that provides clarity. (I've taken Calc 3 and a combined course called "Differential Equations and Linear Algebra," as well as a few other courses.)