Taken from Conway's A Course in Functional Analysis Chapter 3 Section 11 Problem 1:
Problem Statement: Show that $(\mathcal{X}^*)^{**}$ and $(\mathcal{X}^{**})^*$ are equal.
I wanted to see if my solution below was complete.
We show that these two sets include each other. Let our field be $\mathbb{F}$. Suppose that $x \in (\mathcal{X}^*)^{**}$. This is a linear map between linear maps, each from $\mathcal{X} \rightarrow \mathbb{F}$. Since $\mathcal{X}^{**}$ are elements in $\mathcal{X}$, it follows that $\mathcal{X}^*$ are maps from $\mathcal{X}\rightarrow \mathbb{F}$., it follows that $x \in (\mathcal{X}^{**})^*$. Conversely, let $x \in (\mathcal{X}^{**})^*$. $x$ is then a linear functional from $\mathcal{X} \rightarrow \mathbb{F}$. From the description of $ (\mathcal{X}^*)^{**}$ in the previous paragraph, that $x \in (\mathcal{X}^*)^{**}$