Subspaces, Point-Set Topology, Compactness, Matrix Spaces, Limits, Linear Dual Spaces, Hilbert Spaces, Banach Spaces, Subgradients, Gâteaux derivative, Connected Sets, Path Connected Sets, Cauchy Sequences, Power Series, Taylor's Theorem, Properties of Monotonic Functions, Linearity, Hessians, Jacobians, Stone-Weirstrass, Heine-Borel, Fixed Point Theorems, Contraction Principle, Lebesgue Measure, Measurable Functions, L2, Fourier Series....

As an applied mathematician with additional degrees in CS and Engineering, I have used them all. More importantly, the knowledge of those topics helped me understand many algorithms and they lead to a deeper understanding of probability (Lebesgue Measure).