Due September 28
Let a monotone sequence of continuous functions which converges pointwise to the continuous function . Then on .
- Let be a continuous function and define the operator by . Then, the image of the closed ball in under is compact.
Such operator is called a compact operator.
- Let be continuous. Then the operator is compact.
Let an increasing sequence of compact subsets of a metric space, and let . Then there exists p such that .
Let be open and an equicontinuous sequence of functions that converges pointwise. Then converges uniformly on each compact subset of .