Due date: October 20
- If X is discrete, then is discrete.
- Let be a finite set of isolated points of X. Then A is an isolated point in .
Let be nonempty compact sets such that . Then
Two norms are equivalent if there exist constants such that
for all .
- If are equivalent, for all there exist such that and for all , where is the ball with respect to the norm .
- If there exist such that and , then are equivalent.
- If are equivalent, then and are homeomorphic.
- If are equivalent, then is complete if and only if is complete.
- The norms in are equivalent.
- Norms induced by inner products in are equivalent.
- The and uniform norms in satisfy for all , but they are not equivalent between each other.
A rearrangement of the series in a norm space is a series where is a bijection.
- If is absolutely convergent in a Banach space, and , then every rearrangement of converges to x.
- If is conditionally convergent in , then, for all , there exists a rearrangement of that converges to x.