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The identity map for the euclidean norm (i.e. If these laws hold, the space is called Euclidean. is a metric space. In English, we can call these the at-least-as-good-as-x set and the no-better-than-x … 15-11 Completing the Euclidean Plane. (f,f) ≥ 0 and (f,f) = 0 iﬀ f = 0. PROBLEM 1{5. $\begingroup$ If you take a Cauchy sequence and want to prove completeness, you cannot assume that it converges to some point. Euclidean space 5 PROBLEM 1{4. Now we complete the Euclidean plane, by applying the process used to prove the converse part of Theorem 15-28. The Euclidean plane is an affine plane Π' = (P', L'), as it satisfies the axioms (Π'A1), (Π'A2), and (Π'A3). Edit: I am able to prove that a Euclidean space (non-separable) does not need to have a orthonormal basis by using the orthogonal system spanning an infinite Hilbert space as a counter example. Continuous preferences in Euclidean space Assume that the choice set X is a subset of Euclidean n-space