The Space of Continuous Functions
Let k be a natural number, i.e., k is an integer satisfying k ≥ 1. By definition, #x211D; k is the set of all points x = (x1,…, xk where x1…, xk are real numbers. These are called the coordinates of the point x. The point with all coordinates zero is called the origin of ℝ k . For k = 1 the set ℝ k is simply the set ℝ of all real numbers. The set ℝ k is a real vector space with respect to the familiar laws of addition and multiplication by real constants, i.e., if x = (x1,…, x k ), y = (y1,…, y k ) and ⋋ is a real number, then x + y = (x1+y1,…,x k + y k ) and ⋋x = (⋋x1,…, ⋋x k ).
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