One thing I cannot wrap my head around is that there are so many many many conditions for different function spaces, how can you quickly determine which function space a vector/function belongs to?
I think there are 11 conditions for vectors space alone, Hilbert space adds one more, Banach space adds two more, Sobolev space adds....how many more I've lost count.
To know if a function belongs to a certain space, you will have to prove it satisfies each and every single condition for that function space. Even for a function simple as $f(x) = x$, you'd have to prove it for 11 + conditions and how can you ever remember all these conditions?
How do you remember all these conditions and is there a good way to know exactly which function space that a vector/function belongs to upon a glance?