The maximal ideal space (or character space) of a commutative complex Banach algebra 
is the space 
of all characters on
. Thus its points are the
nonzero complex-linear
functionals that are multiplicative. Equipped with the weak*-topology
inherited from the dual normed space 
, it is a locally compact Hausdorff space. This is the topology
of pointwise convergence on 
, so every evaluation map 
for fixed 
is continuous.
If 
is nonzero and has a multiplicative
identity, then 
is nonempty and compact. In this case, the map 
is a bijection
from 
onto the set of maximal ideals
of 
(Taylor, pp. 6-7). For each 
, its Gelfand
transform is the function on 
given by 
.
