If you create a Data Area for a Fabric Lakehouse Prepare Instance with a space in the name, you will discover this is not allowed (because a schema with the same name is created automatically). If you then change the Data Area name, the schema name is unaffected.
It would be useful to have TDI pop up a question asking whether you want the schema name to be changed as well. Similar to how name changes for Conditional Lookups are suggested.