Uniform Convergence Proof

--- Prove: If fn converges to f uniformly and gn converges to g uniformly, then converges uniformly to f/g. --- See attached file for full problem description.

Uniform Convergence of arctan(nx)

Discuss the uniform convergence of the sequence of functions (see the attached file for the function)

Uniform Convergence

(See attached file for full problem description and equations) --- Prove the following theorem. Let be continuous functions on a closed bounded interval . Then uniformly on if and only if for every such that . ---

Find example of uniformly convergent series/Weierstrass test

(See attached file for full problem description and equations) --- Find an example of a sequence of continuous functions on on [0,1] such that the series converges uniformly on [0,1] but the series diverges. Is it a counterexample for the Weierstrass test? ---

Uniform Convergence of series

(See attached file for full problem description and equations) --- Prove the following theorem: If for all and every , and the series converges uniformly in [a,b], then converges uniformly in [a,b]. ---

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 ? - ...continues

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 ...continues

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 tha ...continues