Last update: 18 March 2014
be an open covering of a compact metric space Prove that there exists
such that if then there exists such that
Let be a sequence
in a metric space Prove the following.
converges to if and only if from every subsequence
one can extract a further subsequence converging to
where for all
then no convergent subsequence can exist.
Let be complete and assume that, for every from any sequence
once can extract a further subsequence
Then the sequence admits a convergent subsequence.
Consider the function
be the corresponding mollifications, and let
Prove that but
consequence, although pointwise, one cannot use the Lebesgue dominated
convergence theorem to prove that
be a sequence of absolutely
continuous functions such that
Prove that there exists a subsequence
which converges uniformly on the entire real line.
at the point the sequence
there exists a function such that
the derivatives satisfy
for every and a.e.
Consider a sequence of functions
for every Define
Prove that is Lebesgue measurable and
Let be an absolutely continuous function. Prove that
maps sets of Lebesgue measure zero into sets of Lebesgue measure zero.
is a sequence of functions in
prove that there exists a subsequence that converges pointwise for a.e.
Construct a sequence of measurable functions
but, for each the sequence
has no limit.
For every (nonempty) open set and for
prove that the space
is infinite dimensional.
Construct a sequence of functions
Consider the set with the partial ordering if and only if
Let be a continuous, non decreasing function. Show that the set
is a maximal totally ordered subset of Is every maximal totally ordered subset obtained
in this way?
Give a proof of the generalized Hölder inequality: