This question is addressed directly by Griffiths in his “Introduction to Quantum Mechanics”. It’s in a footnote at the beginning of chapter 3, “Formalisms”. B

This question is addressed directly by Griffiths in his “Introduction to Quantum Mechanics”. It’s in a footnote at the beginning of chapter 3, “Formalisms”. Basically he says that when physicists say “Hilbert space”, they’re actually referring to a specific Hilbert space, namely the set of all square integrable functions, sometimes referred to as “L2”.