Earlier quoted context omitted.
If you have to write complete inner product space a hundred times in an article that's pretty annoying tbh
Terrible excuse. Things are read much more than they are written. That’s like using short variable names rather than descriptive ones as they are easier to type. With current IDEs that isn’t even the case and I’m sure word processors could use the same autocomplete. Every writing shortcut adds cognitive load for the reader to remember.
imo, "Hilbert space" is a lot easier to read, you really only have to recognize the guy's name, it's a fairly easily recognizable word. "Complete inner product space" has me look more closely at more words, and it's all extremely boring-looking words so they don't even stand out in a block of text.