Understanding the second dual space

Question

I am trying to understand the second dual space and the notation in my book. The book Kreyszig defines the dual space as follows:

We define a functional $g_x$ on $X'$ by choosing a fixed $x \in X$ and setting $$g_x(f)=f(x) $$ $$(f \in X' variable)$$

What I don't understand is how $g_x (f)$ works. Is it the composition of functionals? If so, how does the composition equal $f(x)$ because wouldn't $f(x)$ give a scalar value? Thank you for any help...

Answer 1

You want to define a functional on $X'$. That is, a mapping $X' \to \mathbb F$. For each $f\in X'$ you have to output a scalar. So $g_x$ does the job: you input a $f\in X'$, $g_x(f)$ returns a scalar, which is $f(x)$ (note that $f : X\to \mathbb F$, so $f(x)$ is a scalar).

Note that in the above discussion I am fixing a $x\in X$.