Functional Analysis
Hahn–Banach Theorem
A bounded linear functional on a subspace of a normed space extends to the whole space with the same norm
Open Mapping Theorem (Banach)
A surjective bounded linear map between Banach spaces maps open sets to open sets
Closed Graph Theorem
A linear map between Banach spaces with closed graph is automatically continuous
Uniform Boundedness Principle
A pointwise-bounded family of bounded linear operators on a Banach space is uniformly bounded in operator norm
Banach Fixed Point Theorem
Every contraction on a complete metric space has a unique fixed point, reached by iterating the map
Stone–Weierstrass Theorem
A subalgebra of continuous functions on a compact space that separates points is dense in the sup-norm
Riesz Representation Theorem
Every bounded linear functional on a Hilbert space is inner product with a unique vector
Spectral Radius Formula
The spectral radius equals the limit of the nth root of the operator norm of the nth power (Gelfand's formula)