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Extra resources for Appendix to Frigyes Riesz and Bela Sz. -Nagy Functional Analysis...

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Then the set Ab = (x, y) ∈ R to the set of points on the straight line y = bx in the Euclidean plane. The set of all such lines is the class I A = {Ab : b ∈ R} In this case, � Ab = {(0, 0)} � I 2 − {(0, y) : |y| > 0} Ab = R b∈R I b∈R I 20 APPLIED FUNCTIONAL ANALYSIS That is, the only point common to all members of the class is the origin (0, 0), and the union of all such lines is the entire plane R I 2 , excluding the y-axis, except the origin, since b = ∞ ∈ / R. I De Morgan’s Laws can be generalized to the case of unions and intersections of arbitrary (in particular infinite) classes of sets: A− A− � B= � B= B∈B � (A − B) � (A − B) B∈B B∈B B∈B When the universal set U is taken for A, we may use the notion of the complement of a set and write De Morgan’s Laws in the more concise form � � � B B∈B � B B∈B �� = �� = � B� � B� B∈B B∈B De Morgan’s Laws express a duality effect between the notions of union and intersection of sets, and sometimes they are called the duality laws.

1 Let R I be the real numbers and consider the relation R = {(x, y) : x, y ∈ R, I x2 + � y �2 2 = 1} Obviously, R defines the points on the ellipse shown in Fig. 7 (a). Notably, R is not a function, since elements x ∈ R I are associated with pairs of elements in R. I For example, both (0, +2) and (0, −2) ∈ R. 2 The relation R = {(x, y) : x, y ∈ R, I y = sin x} is shown in Fig. 7 (b). This relation is a function. Its domain is R, I the entire x-axis, −∞ < x < ∞. , the y-axis. Its range is the set {y : y ∈ R, I −1 ≤ y ≤ 1}.

Numerous other examples can be cited. 1 Let C denote the set of male children living in a certain residential area. We may introduce a relation “is a brother of” on C × C. If we accept that a given male child can be the brother of himself, then this relation on C is reflexive. Also, if a is the brother of b, then, of course, b is the brother of a. Moreover, if b is also the brother of c, then so is a. It follows that “is the brother of” is an equivalence relation on C. 2 Let L denote the set of all straight lines in the Euclidean plane.

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