Due August 31
Let be Riemann-integrable and .
- f is Riemann-integrable on . (Hint: Given , find a partition of such that where are the upper and lower sums of f with respect to , respectively.)
Consider the functions
Then , but .
Let , a k-continuously differentiable periodic function, with period , and let be its nth Fourier coefficient.
- There exists such that .
- The series converges uniformly if .
Let X be a metric space and its completion. Then, for each , there exists a sequence in X such that in .