ABSTRACT
The Euclidean space https://www.w3.org/1998/Math/MathML"> F m https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315195865/b46e112d-cbc8-4dce-b9d6-367bdebe9455/content/eq1196.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> has the very pleasant property, expressed by the Bolzano–Weierstrass Theorem, that every bounded sequence has a convergent subsequence: https://www.w3.org/1998/Math/MathML"> ( x n ) 1 ∞ ⊆ F m : sup n ∈ ℕ ‖ x n ‖ ≤ C } ⇒ ∃ ( x n k ) 1 ∞ and x ∈ F m : x n k → x as k → ∞ . https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315195865/b46e112d-cbc8-4dce-b9d6-367bdebe9455/content/eq1197.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
