For a Banach space X, let
B(X) be the space of all bounded linear operators on X, and 𝒞 the space of all
compact linear operators on X. In general, the norm-pre-serving extension of a linear
functional in the Hahn-Banach theorem is highly non-unique. The principal result of
this paper is that, for X = c0 or lp with 1 < p < ∞, each bounded linear functional
on 𝒞 has a unique norm-preserving to B(X). This is proved by using a decomposition
theorem for B(X)∗, which takes on a special form for X = c0 or lp with
1 < p < ∞.