Space Elevator in Fiction
While the building of the Space Elevator and its use is in the future many writers of science fiction have used the concept in their works. Science fiction often drives science fact.
Notable works are The Fountains of Paradise by Isaac Asimov, a populariser of science and a promoter of geo-stationary satellites. Kim Stanley Robinson's Mars trilogy which is a series of award-winning science fiction novels that chronicles the settlement and terraforming of the planet Mars through the intensely personal and detailed viewpoints of a wide variety of characters spanning almost two centuries. To connect Mars more closely with Earth a space elevator named "Clarke" is built. Clarke refers to the name of a geosynchronous asteroid to which a space elevator cable is tethered.
A list of the space elevators in fiction is maintained on the general knowledge wikipedia see here