  For a bounded function $f$ from the unit sphere of a closed subspace $X$ of a
Banach space $Y$, we study when the closed convex hull of its spatial numerical
range $W(f)$ is equal to its intrinsic numerical range $V(f)$. We show that for
every infinite-dimensional Banach space $X$ there is a superspace $Y$ and a
bounded linear operator $T:X\longrightarrow Y$ such that $\bar{co} W(T)\neq
V(T)$. We also show that, up to renormig, for every non-reflexive Banach space
$Y$, one can find a closed subspace $X$ and a bounded linear operator $T\in
L(X,Y)$ such that $\bar{co} W(T)\neq V(T)$.
  Finally, we introduce a sufficient condition for the closed convex hull of
the spatial numerical range to be equal to the intrinsic numerical range, which
we call the Bishop-Phelps-Bollobas property, and which is weaker than the
uniform smoothness and the finite-dimensionality. We characterize strong
subdifferentiability and uniform smoothness in terms of this property.
