Definition according to Wikipedia: A shopping mall or shopping centre is a building or set of buildings that contain retail units, with interconnecting walkways enabling visitors to easily walk from unit to unit.
My first mall was in Century City. It was fairly charming outdoor mall when it first opened up. While still an open air shopping center it has grown tremendously since the early days. There was one main department store, Broadway, in the center with wonderful small shops all around. Now there is a major theater complex, multiple major department store, far more shops & restaurants. Century City has grown incredibly, the shopping center is now surrounded by tall building, hotels, & gated communities.
The first enclosed mall I can remember is the Northridge Mall.