The dual space of X is defined to be the space of all linear and continuous functionals that map X to R. But, What exactly is a dual space intuitively?
In my current self-guided understanding, I think of a space of function as a set of points( or a region) in infinite dimensional space $\mathbb R^\infty$. Let $f(x)$ be a element of a space of functions $X$, can I think of each value $f(x)$ as the magnitude in the dimension $x$?
If my assumption above is correct, then what does it mean to have a space consists of functionals? Functionals take a function as input and spit out a scalar, right? There are many functionals that involve differentiation and are not continuous. These functionals in no sense correspond to any functions, right?
Since all linear functionals that are bounded are also continuous, can I say that the only class of functionals that is linear and continuous is simple convolution with certain bounded function g(x)? Namely, $\int f(x)g(x)dx$? And so, all g(x) that make the integral mapping continuous are the elements of the dual space? This is the best explanation I can come up with so far.
If all my assumptions are incorrect, can someone explain to me what it means to have a space which consists of functionals?