Some general (imprecise) notes:
- Function spaces in the left two columns are defined by regularity conditions
- Those in the right three columns are defined by integrability conditions (BMO, the space of functions with bounded mean oscillation, is the dual of the Hardy space )
- Sobolev spaces are defined by some mix of integrability and regularity conditions.
- Most of the spaces above are Banach spaces; some (e.g. ) are Hilbert spaces.
- Functions in any of the above spaces are (a fortiori) measurable.
- Sobolev spaces embed into / Hoelder and spaces; the precise statements can get tricky.
- This map is in no way comprehensive, it only sketches some links between some common families.