Category

Compactness theorems

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Bolzano–Weierstrass theorem

theorem about convergence in a finite-dimensional Euclidean space

Heine–Borel theorem

theorem about compact sets in Euclidean space

Banach–Alaoglu theorem

Arzelà–Ascoli theorem

Cantor's intersection theorem

theorem that a decreasing nested sequences of nonempty closed compact sets has nonempty intersection

Montel's theorem

Prokhorov's theorem

relates tightness of measures to relative compactness in the space of probability measures

Sobolev inequality

theorem about inclusions between Sobolev spaces: if 1≤r≤s<∞ and 1/r − k/n = 1/s − l/n, then Wᵏʳ(ℝⁿ) ⊆ Wˡˢ(ℝⁿ)

Eberlein–Šmulian theorem

Relates three different kinds of weak compactness in a Banach space

On strongly convergent combinations of weakly convergent sequence in a Banach space

Helly's selection theorem

On convergent subsequences of functions that are locally of bounded total variation