Let $X$ be a locally convex topological vector space, let $L$ be a linear subspace of $X$ with the corresponding induced topology, and let $l$ be a continuous linear functional defined over $L$. Then $l$ can be extended to a continuous linear functional over $X$.
Proof: Since $l$ is continuous on $L$, there exists an open barrel set $M$ s.t. $|l(x)|\leq 1$ on $M\cap L$. Let $p$ be the gauge function of $M$. Then $|l|$ is dominated by $p$ on $L$, i.e., $|l(x)|\leq p(x)$ for all $p$ in $L$.
I cannot see why $|l|$ is dominated by $p$ on $L$. Thank you for any help!
Suppose $t>0$ and $\frac 1 t x \in M$. Then $|l(\frac 1 t x) | \leq 1$ so $|l(x)| \leq t$. Taking infimum over all such $t$ we get (by definition of the gauge function) $|l(x)| \leq p(x)$.