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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.

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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.

Keep in mind here that dual spaces require linearity, so a norm is not a linear functional for instance.

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  • $\begingroup$ Hi Ian. Thank you for the help. 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... $\endgroup$ – MathIsHard Jul 11 '18 at 6:09
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    $\begingroup$ @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'. $\endgroup$ – Kavi Rama Murthy Jul 11 '18 at 6:25

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