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?
On
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.
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.