  In 1992, Kiendi, Adamy and Stelzner investigated under which conditions a
certain type of function constituted a Lyapunov function for some
time-invariant linear system. Six years later, it was obtained that this
property holds if and only if the Banach space enjoys the self-extension
property. However, the knowledge of these spaces needed to be extended in order
to make useful this characterization, since there were little information on
which classic Banach spaces satisfy this property and its relations with other
classic properties. We present the self-extension property in a wider frame,
relating it to other well-known spaces as the $1$-injective or $1$-projective
ones. We investigate the property in two important low-dimensional classic
Banach spaces: $\mathbb{R}_1^3$ and $\mathbb{R}_1^4$. We introduce the concept
of $k-$self-extensible spaces and a discussion of the stability of the
property. We also show that every real Banach space of dimension greater than
or equal to $3$ can be equivalently renormed to fail the self-extension
property. Finally, we summarize some consequences of our study and mention some
open questions which appear naturally.
