Basis of a function space.

Orthonormal basis.

Plancherel theorem to approximate functions.

Sobolev embedding theorem to provide a trade-off between integrability and regularity.

Continuous linear functional.

Dual space and adjoint operator.

Young’s inequality for bracketing function spaces.

Galois groups.

The splitting field.

“Given a polynomial function f with rational coefficients, the splitting field of f is defined to be the smallest field that contains all rational numbers and all roots of f.”

“The Galois group of f is the group of all automorphisms of the splitting field.”

The Galois group can be used to show that not all polynomials are solvable by radicals.

Galois groups play a central role in modern algebraic number theory.