Math 516
Professor Lieberman
February 22, 2005
MIDTERM EXAM, corrected version
Directions: You must prepare your solution in LATEX; submit the TEX file via email and turn in
a printed version as well. All references to results (such as theorems, propositions, corollaries, and
lemmata) must be done following the numbering system from class INCLUDING results from Math
515. The statements of these results are available on the web (through the course webpage for Math
516 and at www.public.iastate.edu/˜lieb/fall04/515/515thm.pdf for Math 515). The solutions to
these problems will be graded on form as well as content. If you have any questions, ask me. Note
that there are only two questions on this test.
This exam is due at 4:00 pm, Monday, March 7.
1. Let X be a Banach space and let T be a bounded linear operator on X with kT k < 1.
Prove that the geometric series 1 + T + T 2 + · · · + T n + . . . converges in X ∗ . What does it
converge to?
2. Prove that a subset of Rn is compact if and only if it is bounded and closed.
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