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FA 2017/5/9 Both sides of the answer sheet can be used.

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FA 2017/5/9 Both sides of the answer sheet can be used.

1 Describe the Banach’s fixed-point-theorem on contractions and give an existence theorem which can be handled with it.

2 Give the definition of the sequential Banach space

p

for 1 p < and show that it contains a countable dense subset.

3 Let a and b be positive reals. For integrable functions f and g on the real line

R

satisfying f(t) = 0 for | t | > a and g(t) = 0 for | t | > b, show that (f g)(t) = 0 for | t | > a + b.

1

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