It is a result of Holub’s [Math. Ann. 201 (1973), 157-163], that for T a compact operator on a real Hilbert space, T is smooth
⇔
‖
T
x
1
‖
=
‖
T
x
2
‖
=
‖
T
‖
\Leftrightarrow \left \| {T{x_1}} \right \| = \left \| {T{x_2}} \right \| = \left \| T \right \|
for some
‖
x
1
‖
=
‖
x
2
‖
=
1
\left \| {{x_1}} \right \| = \left \| {{x_2}} \right \| = 1
implies
x
1
=
±
x
2
{x_1} = \pm \;{x_2}
. We extend this characterization of smooth, compact operators to a large class of Banach spaces, including
l
p
,
L
p
[
0
,
1
]
{l_p},{L_p}[0,1]
, and
d
(
a
,
p
)
d(a,p)
, with
1
>
p
>
∞
1 > p > \infty
. We show that for this same class of Banach spaces, one dimensional, norm one functionals in
K
(
X
)
∗
K{(X)^\ast }
must be extremal. We also present examples of spaces for which Holub’s condition does not characterize smooth, compact operators.