= Solution
For a <normed space> $X$, let $X^*$ be its <continuous dual space> and $X^{**}=(X^*)^*$ its <bidual space>. Its <canonical embedding into the bidual> is
$$
J_X:X\to X^{**},\qquad(J_Xx)(\ell)=\ell(x).
$$
The definition of a <reflexive space> in this setting is
$$
\boxed{J_X(X)=X^{**}.}
$$
The <Hahn-Banach theorem> makes $J_X$ a <linear map> that is an <isometry>. For any <normed space> $Z$, its <continuous dual space> is a <Banach space>: a <Cauchy sequence> $(\ell_n)$ in the <operator norm> has scalar limits $\ell(z)=\lim_n\ell_n(z)$, and the uniform Cauchy bound proves both boundedness of $\ell$ and $\|\ell_n-\ell\|\to0$. Apply this with $Z=X^*$ to see that $X^{**}$ is <complete>. If $J_X$ is onto, the <isometry> transfers this <completeness> to $X$. Therefore
$$
\boxed{X\text{ reflexive}\ \Longrightarrow\ X\text{ complete}.}
$$
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