Analytic Functions – nice but sparse. Examples include: exp, sin, polynomials.
Test functions. Infinitely differentiable on closed interval [0,1].
Continuous functions. Good for evaluation defined on [0,1].
Square integrable functions.
Borel measures.
Distributions.
Hyperfunctions.
Distributions as a weak limit of test functions.
Distributions are useful in the study of partial differential equations.