  Let $X$, $Y$ be two real Banach spaces and $\varepsilon>0$. A standard
$\varepsilon$-isometry $f:X\rightarrow Y$ is said to be
$(\alpha,\gamma)$-stable (with respect to $T:L(f)\equiv\overline{{\rm
span}}f(X)\rightarrow X$ for some $\alpha, \gamma>0$) if $T$ is a linear
operator with $\|T\|\leq\alpha$ so that $Tf-Id$ is uniformly bounded by
$\gamma\varepsilon$ on $X$. The pair $(X,Y)$ is said to be stable if every
standard $\varepsilon$-isometry $f:X\rightarrow Y$ is $(\alpha,\gamma)$-stable
for some $\alpha,\gamma>0$. $X (Y)$ is said to be universally left
(right)-stable, if $(X,Y)$ is always stable for every $Y (X)$. In this paper,
we show that universal right-stability spaces are just Hilbert spaces; every
injective space is universally left-stable; a Banach space $X$ isomorphic to a
subspace of $\ell_\infty$ is universally left-stable if and only if it is
isomorphic to $\ell_\infty$; and that a separable space $X$ satisfies the
condition that $(X,Y)$ is left-stable for every separable $Y$ if and only if it
is isomorphic to $c_0$.
