= Solution
The <continuous dual space> $X'$ is the <vector space> of bounded scalar-valued <linear functionals>, with <operator norm>
$$
\boxed{\|f\|=\sup_{\|x\|\leq1}|f(x)|.}
$$
This makes it a normed space, and in fact a <Banach space> whether or not $X$ is complete.
The <norm>-preserving <Hahn-Banach theorem> states that a <bounded linear functional> $f_0$ on any <linear subspace> $M\subseteq X$ extends to a <bounded linear functional> $f$ on all of $X$ with $f|_M=f_0$ and $\|f\|=\|f_0\|$. The subspace need not be closed. This holds over both the real and complex scalar fields, using the complex version when necessary.
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