Let $(E,\Vert.\Vert)$ be a real Banach space and $\ell\ne 0$
an non-continuous linear form on $E$.
Let $a\in E$ be such that $\ell(a)=1$.
König-Wittstock [Non-equivalent complete norms
and would-be continuous linear functionals,
Expositiones Mathematicae, 10 (1992), 285--288]
define a new norm on $E$ by
$$
\left\vert\mkern -1.5mu\Vert\right.
x
\left\vert\mkern -1.5mu\Vert\right.
=\vert\ell(x)\vert + \inf_{t\in \mathbb R} \Vert x-ta\Vert.
$$
I don't know how to prove that $(E,\left\vert\mkern -1.5mu\Vert\right.{.}\left\vert\mkern -1.5mu\Vert\right.)$
is also a Banach space.
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$\begingroup$ Your space is isometric (by the map $x \in E \mapsto (\ell(x), x+\mathbf R a)$) to the $\ell_1$-direct sum of the Banach spaces $\mathbf R$ and $E/\mathbf R a$. $\endgroup$– Mikael de la SalleJul 22, 2014 at 13:10
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