  Let $(A, \|\cdot\|)$ be any normed algebra (not necessarily complete nor
unital). Let $a \in A$ and let $V_A(a)$ denote the spatial numerical range of
$a$ in $(A, \|\cdot\|)$. Let $A_e = A + {\mathbb C} 1$ be the unitization of
$A$. If $A$ is faithful, then we get two norms on $A_e$; namely, the operator
norm $\|\cdot\|_{op}$ and the $\ell^1$-norm $\|\cdot\|_1$. Let $A^{op} = (A,
\|\cdot\|_{op})$, $A_e^{op} = (A_e, \|\cdot\|_{op})$, and $A_e^1 = (A_e,
\|\cdot\|_1)$. We can calculate the spatial numerical range of $a$ in all these
three normed algebras. Because the spatial numerical range highly depend on the
identity as well as on the completeness and the regularity of the norm, they
are different. In this paper, we study the relations among them. Most of the
results proved in \cite{BoDu:71, BoDu:73} will become corollaries of our
results. We shall also show that the completeness and regularity of the norm is
not required in \cite[Theorem 2.3]{GaHu:89}.
