By Hugo D. Junghenn

ISBN-10: 148221928X

ISBN-13: 9781482219289

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3 hold in the extended system R. 14 Definition. An interval in R is a nonempty set I with the property that a, b ∈ I and a < x < b imply that x ∈ I. An interval containing more than one point is said to be nondegenerate. ♦ Arguing cases, one may show that the definition of interval reduces to the following familiar subsets of R: (a, b) := {x : a < x < b}, (a, b] := {x : a < x ≤ b}, [a, b) := {x : a ≤ x < b}, [a, b] := {x : a ≤ x ≤ b}. For example, if I is unbounded below and bounded above with b := sup I ∈ I, then I = (−∞, b].

9}. The representation n = dp dp−1 . . d0 is called the decimal positional notation for n. 6 Euclidean Space The real number system may be used to construct other important mathematical systems, such as n-dimensional Euclidean space and the complex number system. In this section we construct the former. The reader may delay reading this section, as the material will not be needed until Chapter 8. For n ∈ N, let Rn denote the set of all n-tuples x := (x1 , x2 , . . , xn ), where xj ∈ R. Each such n-tuple is called a point or vector, depending on context.

3 Theorem. The inner product in Rn has the following properties: (a) x · x = x 22 . (b) x · y = y · x (commutativity). (c) t(x · y) = (tx) · y = x · (ty) (associativity). (d) x · (y + z) = (x · y) + (x · z) (additivity). (e) |x · y| ≤ x 2 y 2 (Cauchy–Schwartz inequality). Proof. Properties (a) and (b) are immediate and parts (c) and (d) follow respectively from the calculations n n xj yj = t j=1 n n (txj )yj = j=1 xj (tyj ) and j=1 n xj (yj + zj ) = j=1 n xj yj + j=1 xj zj . j=1 The inequality in (e) holds trivially if y = 0.

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