This inspired me to compile a list:
Since, I’m not a mathematician(pure/applied) I just compiled things from the blog post combining
with the comments:
* Bayes Theorem

• Jensen’s Inequality if is a convex function and X is
  a random variable. Extends convexity from sums to integrals(aka discrete to continuous)
• lto’s lemma: aka(Merton, Black and Scholes option pricing formula)
• Complex analysis.. should i disqualify this as not a theorem??
• Standard error of the mean.details link
• Jordan Curve Theorem: A closed curve has an inside and an outside. (sounds obvious in 2D
  and 3D, perhaps with time as 4D, keeping options open is staying outside closed curves??)
• kullback-leibler positivity:(no clue need to look up wolfram alpha or wikipedia)
• Hahn-Banach Theorem (again needs searching)
• Pigeon-Hole principle link here
• Taylor’s theorem, (once again continuous function approximated by sum of discrete
  components/expressions) Used in:
• Approximating any function with nth degree precision
• Bounding the error term of an approximation
• Decomposing functions into linear combinations of other functions
• Kolmogorov’s Inequality for the maximum absolute value of the partial sums of a sequence of IID random variables.( the basis of martingale theory)
• Karush-Kuhn-Tucker optimality conditions for nonlinear programming, link
  here
• Envelope Theorem — from economics
• Zorn’s lemma , also Axiom of Choice
• Fourier Transform and Fast Fourier Transform