  In this paper we study two properties viz. property-$U$ and property-$SU$ of
a subspace $Y$ of a Banach space which correspond to the uniqueness of the
Hahn-Banach extension of each linear functional in $Y^*$ and in addition to
that this association forms a linear operator of norm-1 from $Y^*$ to $X^*$. It
is proved that, under certain geometric assumptions on $X, Y, Z$ these
properties are stable with respect to the injective tensor product; $Y$ has
property-$U$ ($SU$) in $Z$ if and only if $X\otimes_\e^\vee Y$ has property-$U$
($SU$) in $X\otimes_\e^\vee Z$. We prove that when $X^*$ has the
Radon-Nikod$\acute{y}$m Property for $1<p< \infty$, $L_p(\mu, Y)$ has
property-$U$ (property-$SU$) in $L_p(\mu, X)$ if and only if $Y$ is so in $X$.
We show that if $Z\subseteq Y\subseteq X$, where $Y$ has property-$U$ ($SU$) in
$X$ then $Y/Z$ has property-$U$ ($SU$) in $X/Z$. On the other hand $Y$ has
property-$SU$ in $X$ if $Y/Z$ has property-$SU$ in $X/Z$ and $Z (\subseteq Y)$
is an M-ideal in $X$. It is observed that a smooth Banach space of dimension
$>3$ is a Hilbert space if and only if for any two subspaces $Y, Z$ with
property-$SU$ in $X$, $Y+Z$ has property-$SU$ in $X$ whenever $Y+Z$ is closed.
We characterize all hyperplanes in $c_0$ which have property-$SU$.
