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Rudin Errata

# Rudin Errata - -1ERRATA AND ADDENDA TO CHAPTERS 1-7 OF...

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-1- ERRATA AND ADDENDA TO CHAPTERS 1-7 OF RUDIN’S PRINCIPLES OF MATHEMATICAL ANALYSIS , 3rd Edition, 4th Printing (noted as of August, 2003) For additional errata to earlier printings , see last page of these sheets. Note: If you don’t want to write corrections into your text, you might put them on PostIts (or slivers of paper cut from PostIts) and insert these at the page in question. P.4, line 4: Change this line to ‘‘(ii) If γ S is an upper bound of E , then α ’’ for greater clarity P.4 3rd line of Definition 1.10: A clearer statement would be, ‘‘Every subset E S which is nonempty and bounded above has a supremum sup E in S .’’ P.5, last 5 lines of proof of Theorem 1.11 : Change these lines to: If were not a lower bound of B , there would be some x B satisfying x < . This x would be an upper bound of L (by the preceding paragraph), contradicting our assumption that is the least upper bound of L .S o is a lower bound of B . Now if y is any lower bound of B , then y L ,s o y sup L = ; this shows that is the greatest lower bound of B . P.6, Proposition 1.14: Add ( e )– ( x+y )=(– x )+(– y ). Can you see how to prove this? I will either discuss it in class, or make it an exercise. P.12, definitions of operations on the extended real numbers: Rudin should have noted the convention that x +(+ ) and x +(– ) may be abbreviated x + and x respectively, and mentioned that addition and multiplication are understood to be commutative on the extended reals, so that the definitions he gives also imply further cases like + + x =+ . Finally, the three equations in ( a ), instead of having the common condition ‘‘If x is real’’, should be preceded by the respective conditions, ‘‘If x is real or + ’’, ‘‘If x is real or – ’’, and only in the last case simply ‘‘If x is real’’. P.16, Theorem 1.37: Add one more part: ( g ) Assuming k > 0 , there exists a vector u with | u |=1 such that u·x =| x |. Proof: If x 0 let u x | –1 x ;i f x = 0 let u be any vector with | u . P.19, middle: The author refers to the archimedean property of Q . This is not a consequence of Theorem 1.20(a); that would be circular reasoning. Rather, it is an elementary property of Q : Given x, y Q with x > 0, we need to find an n>y∕x .I f y∕x < 0, take n = 1; otherwise, write as a fraction with positive denominator, and take for n any integer greater than its numerator. P.36: After finishing the section of metric spaces, you might find the following discussion enlightening; but it is not required reading. What is topology? Chapter 2 of Rudin is entitled ‘‘Basic Topology’’, but the chapter is about metric spaces, and the word ‘‘topology’’ does not appear in that chapter, nor in the index. What does it refer to? Topology is a field of mathematics that includes the study of metric spaces as a special case. The key to the connection between metric spaces and the more general concept of a topological space is Theorem 2.24, parts (a) and (b) (p.34), which show that if we write T for the set of all open sets in a metric space, then the union of any

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Rudin Errata - -1ERRATA AND ADDENDA TO CHAPTERS 1-7 OF...

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