By John B. Conway
This booklet is an introductory textual content in sensible research. in contrast to many sleek remedies, it starts off with the actual and works its method to the extra common. From the studies: "This booklet is a superb textual content for a primary graduate direction in useful analysis....Many attention-grabbing and significant functions are included....It comprises an abundance of workouts, and is written within the attractive and lucid sort which we've got come to anticipate from the author." --MATHEMATICAL studies
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Extra resources for A Course in Functional Analysis (Graduate Texts in Mathematics, Volume 96)
5m42 Ϫ 20 108. 2x 2 Ϫ 288 3 2 ͑ y Ϫ 3͒ ͑ y ϩ 2͒ ͑ y Ϫ 2͒ ͑x Ϫ 1͒ ͑x ϩ 1͒ 3 2 ͑ p Ϫ 2͒ ͑ p ϩ 3͒ ͑ p Ϫ 3͒ ͑x ϩ 1͒ ͑x Ϫ 1͒ 2 5͑m ϩ 2͒ ͑m Ϫ 2͒ 2 2͑x ϩ 12͒ ͑x Ϫ 12͒ 2 109. 2x 3 ϩ 6x 2 Ϫ 8x Ϫ 24 110. 3x 3 ϩ 6x 2 Ϫ 27x Ϫ 54 2͑x ϩ 3͒ ͑x ϩ 2͒ ͑x Ϫ 2͒ 3͑x ϩ 2͒ ͑x ϩ 3͒ ͑x Ϫ 3͒ 111. 4c 2 Ϫ 4cd ϩ d 2 112. 3 m6 ϩ 8m33 Ϫ 20 114. x 4 Ϫ 37x 2 ϩ 36 ͑2c Ϫ d͒ 2 ͑m ϩ 10͒ ͑m Ϫ 2͒ 115. p Ϫ 64p4 ͑3a Ϫ b͒ 2 ͑x ϩ 1͒ ͑x Ϫ 1͒ ͑x ϩ 6͒ ͑x Ϫ 6͒ 116. 125a Ϫ 8a4 p͑1 Ϫ 4p͒ ͑1 ϩ 4p ϩ 16p ͒ 2 2 118. Explain how the rule for factoring a sum of cubes can be used to factor a difference of cubes.
4. ͱ n n a ͙a n b ͙b (b 0). 5. ͙ am ͑ ͙ a ͒ . n n EXAMPLE 2 m Simplify each of the following. 6 f ) ͙ 85 ͙͑ 8 ͒ 25 32 3 5 3 41 • Radical Notation and Rational Exponents Using Property 5 g) ͙216x 5y 3 ͙36 и 6 и x 4 и x и y 2 и y Using Property 3 ͙36x 4y 2 ͙6xy ͉6x 2y͉͙6xy Using Property 1 2 6x ͉ y͉͙6xy 6x2 cannot be negative, so absoluteh) ͱ value signs are not needed for it. 2 ͙x 2 x 16 ͙16 ͉x͉ 4 Using Property 4 Using Property 1 In many situations, radicands are never formed by raising negative quantities to even powers.
EXAMPLE 1 Factor each of the following. a) 15 ϩ 10x Ϫ 5x 2 b) 12x 2y 2 Ϫ 20x 3y Solution a) 15 ϩ 10x Ϫ 5x 2 5 и 3 ϩ 5 и 2x Ϫ 5 и x 2 5͑3 ϩ 2x Ϫ x 2 ͒ We can always check a factorization by multiplying: 5͑3 ϩ 2x Ϫ x 2 ͒ 15 ϩ 10x Ϫ 5x 2. b) There are several factors common to the terms of 12x 2y 2 Ϫ 20x 3y , but 4x 2y is the “largest” of these. 12x 2y 2 Ϫ 20x 3y 4x 2y и 3y Ϫ 4x 2y и 5x 4x 2y͑3y Ϫ 5x͒ Factoring by Grouping In some polynomials, pairs of terms have a common binomial factor that can be removed in a process called factoring by grouping.