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There is something basic I don't understand about the second dual space. Assume $X$ is a normed linear space. Denote by $X^{*}$ the dual space of $X$. If we fix an $x \in X$, $X^{*}(x)$ is a bounded linear functional and their space is called the second dual.

Here comes the part I don't understand. Isn't the second dual just $X$ again? What am I missing?

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    $\begingroup$ If $X^*$ is a space, what is $X*(x)$. Do you mean $f \in X^*$ and then $f(x)$? Because then $f(x) \in \mathbb R$ or $\mathbb C$. $\endgroup$
    – Ramanujan
    Apr 20 '20 at 8:24
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Spaces $X$ for which $X^{**}=X$ are called reflexive spaces. Not every Banach space is reflexive. (An incomplete normed linear space can never be reflexive). $C[0,1]$ is an example of a non-reflexive space. All finite dimensional spaces are reflexive.

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I think your problem is that you mixed the "continuous" dual and the "algebraic" dual. For a given normed vector space $X$ over $\mathbb{R} $ or $\mathbb{C} $, we have the dual $X^{\ast } $ consisting of all continuous linear functionals on $X$, and the "dual space" $X'$ consisting of all linear functionals on $X$. For the second one, we naturally have $X''\cong X$ algebraically as vector spaces(this is a wrong argument. In fact, we have $\dim X'\geq 2^{\dim X} $. See page 207 of Hungerford's Algebra.), but for the first one, there is no reason that the second dual is isomorphic to $X$. The answer of Murthy has given a counter-example.

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  • $\begingroup$ You hit the nail on the head. I was mixing the continuous and the Algebric dual! $\endgroup$
    – daniel
    Apr 20 '20 at 8:41
  • $\begingroup$ In general, we only have $X\hookrightarrow X''$ with the algebraic dual as well, they are not isomorphic if $\dim X\ge\omega$, are they? $\endgroup$
    – Berci
    Apr 20 '20 at 11:05
  • $\begingroup$ @Berci You are right. Hungerford's Algebra gives the result that a dual of a vector space with dimension infinity would have $\dim X'>\dim X$ as cardinal numbers.(Exactly, exercise 12 in page 207.) $\endgroup$
    – TheWildCat
    Apr 21 '20 at 6:25

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