Mazur-Ulam Theorem

The notation and terminology used in this paper have been introduced in the following papers: [19], [18], [4], [5], [20], [11], [10], [14], [17], [1], [6], [16], [24], [25], [21], [13], [12], [22], [2], [9], [8], [3], and [7]. For simplicity, we use the following convention: E, F , G are real normed spaces, f is a function from E into F , g is a function from F into G, a, b are points of E, and t is a real number. Let us note that I is closed. Next we state four propositions: (1) DYADIC is a dense subset of I. (2) DYADIC = [0, 1]. (3) a+ a = 2 · a. (4) (a+ b)− b = a. Let A be an upper bounded real-membered set and let r be a non negative real number. Observe that r ◦A is upper bounded. Let A be an upper bounded real-membered set and let r be a non positive real number. Note that r ◦A is lower bounded. Let A be a lower bounded real-membered set and let r be a non negative real number. Observe that r ◦A is lower bounded.