# Is the norm an element of the second dual space?

I am wondering if a norm is an example of an element of the second dual space? It takes in functionals and spits out a number.

I am trying to have some intuition for this so I have been reading posts on it but I didn’t see any concrete examples.

What I mean is, I have the following in mind:

We have the field which are numbers/vectors . Ex: 1, (1,2,3)

We have the dual space which are functions. Ex: f(x)=x^2

We have the second dual space which are functionals that take in functionals and create an element of the field. Ex: norms?

Or maybe I am way off here.... :/

Edit: Could an example be a derivative evaluated at a number? The derivative takes in a function and then generates a number in the field? I’m sorry if I’m off again here.... I am struggling to understand how a function can take in a function and create a number.

The canonical elements of the second dual space are essentially the elements of the closure of the original space, specifically they are functions $F[f]=f(x)$ for a fixed $x$. Often, any other elements of the second dual space are quite weird/hard to construct if they exist at all.
• @Ian There are many cases where the where the second dual is very concrete even without reflexivity. For example, the second dual of $c_0$ is $l^{\infty }$ which is not .weird'. – Kavi Rama Murthy Jul 11 '18 at 6:25