They belong to the mystical realm of mathematics, but not only are they powerful, but also their might is conveyed by their names.
  1. Abstract Riesz-Thorin Interpolation Theorem
    The non-abstract version talks about interpolating between Lp spaces and also operators between them. Now extrapolate to the abstract version!
  2. Arzela-Ascoli
    Gives some intuition about compactness of sets of continuous functions.
  3. Banach-Alaoglu
    The closed unit ball of a normed space is weak-*-compact.
  4. Bolzano-Weierstraß
  5. Hadamard Three Line Theorem