# Functional Analysis

solution Let and be metric spaces, and be a mapping.

(i) Prove that if is open whenever is open, then is continuous.

(ii) Prove that is continuous if and only if is closed whenever is closed.

solution (i) Let with maximum metric . Prove that is seperable.

(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 Show that any Cauchy sequence in a metric space is always bounded.

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 .

solution Let be a closed subspace of a normed space . Let but not in . Use the Hahn-Banach Theorem to prove that there exists a bounded linear functional on such that

(a) for every

(b)

(c)

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?