Math 240                              Proof Evaluation #1                              Fall,1998
M. Flashman                                                                              Due:9-16-98

Read the short note "The Kantorovich Inequality" by Vlastimil Pták from The American Mathematical Monthly, V. 102, No. 9, November, 1995, pp. 821, 822.

1. Is the inequality statement conditional or absolute? If it is conditional, what are the hypotheses and conclusions? If it is absolute what is the main quantifier?

2. List the variables used in inequality. Indicate what these variables represent. List any words or symbols used in the proof that were not explained or you didn't understand.

3. Is the proof of the inequality direct or indirect?

• a. If the proof is indirect, state the way in which the argument proceeds.(What is assumed? What is actually demonstrated?)
• b. If the proof is direct, does the proof proceed forward or is it mixed with some backward argument? If it has some backward argument, indicate briefly how the original conclusion is altered.

4. Did the proof explicitly leave some steps for the reader to complete? If so, state what steps the reader is expected to complete. [Optional: Complete these steps.]

5. Did the proof implicitly leave some steps for the reader to complete? If so, state what steps you think the reader is expected to complete. [Optional: Complete these steps.]

6. Indicate any parts of the argument that you felt needed greater detail or better connection to the proof. [Optional: Supply these details or suggest a better connection.]

7. Overall, do you think this proof was effective? Discuss briefly the basis for you conclusion.