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.