Uniform Convergence

(See attached file for full problem description and equations) --- Prove: Let be a sequence of continuous functions convergent uniformly on a bounded closed interval [a,b] and let . For n = 1,2,…., define . Then the sequence converges uniformly on [a,b]. Is the same true if [a,b] is replaced by ? ---

Linear Isometry

(See attached file for full problem description and equations) --- Let be a measurable space and let be two -finite measures defined on . Suppose and is the Radon-Nikodym derivative of with respect to . Define by Show that is a well-defined linear isometry and is an isomorphism if and only if (i.e are mutually absolutely continuous). ---

Integral operator

(See attached file for full problem description with equations) --- If is a measure space and , show that defines a bounded integral operator.

Proof of existence of a one-to-one correspondence between the open interval (0, 1) and the half-open interval (0, 1]

Prove that there is a bijection from the open interval (0, 1) to the half-open interval (0, 1].

Partial order

(See attached file for full problem description with equations) --- Let be a vector space and a subset of such that implies and for Define a partial order on by defining to mean . A linear functional on is said to be positive (with respect to ) if for . Let be any subspace of with the property that for each there is an with . Assume that , where Then each positive linear functional on can be extended to a positive linear functional on . ---