Given a linear control system in a Hilbert space with a bounded control operator, we establish a characterization of exponential stabilizability in terms of an observability inequality. In general, characterizations of stabilization are presented by certain frequency conditions which are in "fequency domain”. Our characterization is given in "time domain". The way to approach the aim is as: we realize that the exponential stabilization is equivalent to a special kind of controllability, and then by the duality argument, it is equivalent to a weak observability inequality.