Trying to understand the idea of a Dual Space and I'm having a little trouble understanding what is meant by the idea of a linear functional. My understanding is that a functional maps from either an n-tuple or a polynomial to some field, but I don't understand how these concepts are supposed to be linear.
Like, if we consider an easier example, say R^2, what would a linear functional look like? Would it have to map to the field of real numbers, R, or any field? How can a functional be linear if you need two variable inputs, like x and y for R^2?
And finally, if we define the Dual Space to be the set of all possible linear functionals from the Vector Space to a field, how exactly is this itself a Vector Space? Are the vectors the functionals themselves? And if so, what is the field over which the Dual Space is a Vector Space?
Thank you in advance for your help.