By Arthur Wouk

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**Extra info for A course of applied functional analysis**

**Sample text**

15 The triangle in Example 8. lengths of the sides of the triangle satisfy the Pythagorean relationship a 2 + b 2 = c2. Applying the distance formula we find that a = 21-3 - 122 + 14 - 022 = 232, b = 211 - 522 + 10 - 422 = 232, c = 21 -3 - 522 + 14 - 422 = 264. The triangle is a right triangle because a 2 + b 2 = 123222 + 123222 = 32 + 32 = 64 = c2. Now try Exercise 39. Properties of geometric figures can sometimes be confirmed using analytic methods such as the midpoint formulas. EXAMPLE 9 Using the Midpoint Formula It is a fact from geometry that the diagonals of a parallelogram bisect each other.

1 66. Decimal Forms of Rational Numbers Here is the third step when we divide 1 by 17. 00 85 15 By convention we say that 1 is the first remainder in the long division process, 10 is the second, and 15 is the third remainder. (a) Continue this long division process until a remainder is repeated, and complete the following table: Real Numbers 11 In Exercises 69–72, solve these problems without using a calculator. 69. Multiple Choice Which of the following inequalities corresponds to the interval 3- 2, 12?

1. - 37/8 2. 15/99 17. x 7 - 3 3. - 13/6 4. 5/37 19. In Exercises 5–10, describe and graph the interval of real numbers. 5. x … 2 6. - 2 … x 6 5 7. 1 - q , 72 8. 3 - 3, 34 10. x is greater than or equal to 2 and less than or equal to 6. In Exercises 11–16, use an inequality to describe the interval of real numbers. 12. 1- q , 44 13. x –5 –4 –3 –2 –1 0 1 2 3 4 x 0 1 2 3 4 0 1 2 3 4 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 20. x 22. x is positive. In Exercises 23–28, use words to describe the interval of real numbers.