(ii) Use (i) to prove that is also seperable with respect to the metric .
solution Assume as . Prove that is continuous when is continuous for all .
solution Let for .
(i) Show that .
(ii) Use (i) to prove that
(iii) Use (ii) to get
solution Use the inequality for to prove that
for and , and then use this to prove the triangular inequality:
solution Assume that and are two equivalent norms on , and . Prove that is compact in if and only if is compact in .
solution Prove the Maximum and Minimum Theorem for continuous functions on compact sets.
solution Let be a closed subspace of a normed space . Define on the quotient space by
for every . Prove that is a norm.
solution Assume that is a countable linearly independent subset of a vector space , and span . Prove that admits two inequivalent norms.
solution Let be defined by
(i) Show that is a bounded linear operator.
(ii) Find .
(iii) Show that is one-to-one but not onto. Find the range space of .
solution Prove that the dual space of is .
solution Let such that . Show that for every .
solution Let and where 1 is in the th position. Define for . Show that for each , but .
solution Let be a bounded linear operator. Show that exists and is bounded if and only if there exists such that for every .
solution Let be a normed space and . Prove that if for every bounded linear functional on , then .
(a) for every
solution Let be a bounded linear operator from a normed space to a normed space , and its norm be defined by
Show that .
solution Let and be two normed spaces and be a linear mapping from to . Show that if is continuous, then the null space is a closed subspace of . Give an example showing that the closedness of does not imply the continuity of .
solution State the Hahn-Banach Extension theorem for bounded linear functionals on normed spaces.
solution Let be a normed space. Use the Hahn-Banach Extension to prove that for every , there exists a bounded linear functional on such that and .
solution State the Uniform Boundedness Theorem.
solution Let and be Banach spaces and . Show that the following are equivalent:
(a) is bounded;
(b) is bounded for each ;
(c) is bounded for all and all .
solution Let and define . Show that
defines a bounded linear operator on when is a continuous function on . Estimate the norm of .
solution Let and where is in the th position. Let be a sequence of complex numbers.
(a) Show that defines a bounded linear operator on if and only if . In this case, find the norm of .
(b) Find the necessary and sufficient condition for to be bounded invertible (i.e., the inverse exists and is bounded).
solution Let be a sequence of real numbers such that is convergent for every . Use the Uniform Boundedness theorem to prove that .
solution Let be a normed space and be a linearly independent subset of . Use the Hahn-Banach extension theorem to show that there exist bounded linear functionals on such that
where when and when . Can you think about extending this to an infinite sequence of vectors?