Functional Analysis

Download Analysis with an introduction to proof. by Steven R. Lay PDF

By Steven R. Lay

via introducing common sense and by means of emphasizing the constitution and nature of the arguments used, this booklet is helping readers transition from computationally orientated arithmetic to summary arithmetic with its emphasis on proofs. makes use of transparent expositions and examples, worthwhile perform difficulties, quite a few drawings, and chosen hints/answers. deals a brand new boxed assessment of keyword phrases after every one part. Rewrites many workouts. gains greater than 250 true/false questions. comprises greater than a hundred perform difficulties. offers exceedingly high quality drawings to demonstrate key principles. presents a number of examples and greater than 1,000 routines. a radical reference for readers who have to raise or brush up on their complicated arithmetic abilities.

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Example text

11. Fill in the blanks in the proof of the following theorem. THEOREM: A ⊆ B iff A ∩ B = A. Proof: Suppose that A ⊆ B. If x ∈ A ∩ B, then clearly x ∈ A. Thus A ∩ B ⊆ A. On the other hand, ______________________________ _______________________________________________________. Thus A ⊆ A ∩ B, and we conclude that A ∩ B = A. Conversely, suppose that A ∩ B = A. If x ∈ A, then __________ ________________________________________________________. Thus A ⊆ B. ♦ 12. Suppose you are to prove that set A is a subset of set B.

On the other hand, ______________________________ _______________________________________________________. Thus A ⊆ A ∩ B, and we conclude that A ∩ B = A. Conversely, suppose that A ∩ B = A. If x ∈ A, then __________ ________________________________________________________. Thus A ⊆ B. ♦ 12. Suppose you are to prove that set A is a subset of set B. Write a reasonable beginning sentence for the proof, and indicate what you would have to show in order to finish the proof. 13. Suppose you are to prove that sets A and B are disjoint.

This might seem to be an unwarranted assumption, but really it is not. If S is the empty set, then of course S ⊆ T, so the only nontrivial case to prove is when S is nonempty. 13(f ). Proof: We wish to prove that A \(B ∪ C ) = (A \ B) ∩ (A \ C ). To this end, let x ∈ A \(B ∪ C ). Then ___________ and ____________. Since x ∉ B ∪ C, __________ and x ∉ C (for if it were in either B or C, then it would be in their union). Thus x ∈ A and x ∉ B and x ∉ C. Hence x ∈ A \ B and x ∈ A \ C , which implies that ____________.

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