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.