Let E be an infinite-dimensional linear subspace of
C
(
S
)
C(S)
, the space of bounded continuous functions on a locally compact Hausdorff space S. If
μ
\mu
is a regular Borel measure on S, then each element of E may be regarded as a multiplication operator on
L
p
(
μ
)
(
1
≦
p
>
∞
)
{L^p}(\mu )(1 \leqq p > \infty )
. Our main result is that the strong operator topology this identification induces on E is properly weaker than the strict topology. For E the space of bounded analytic functions on a plane region G, and
μ
\mu
Lebesgue measure on G, this answers negatively a question raised by Rubel and Shields in [9]. In addition, our methods provide information about the absolutely p-summing properties of the strict topology on subspaces of
C
(
S
)
C(S)
, and the bounded weak star topology on conjugate Banach spaces.