Lecture 3, September 20, 2007

We finish Chapter 2 in R&Y:

If X is a normed space, then it has an associated metric topology, so that now finally the combination of the two basic ingredients of FA (linear structure and topology) has become visible.

One can build new normed spaces from old ones, by considering subspaces, products and quotients by closed subspaces (the latter is not in R&Y). The corresponding norm topologies are then precisely the ones which result from the general constructions in topology.

Two norms are equivalent (as defined in R&Y) precisely when their associated topologies are the same. This result is not in R&Y (but it is in the homework).

We treat the basic material on finite dimensional normed spaces: all norms are equivalent and the space is always complete. If a subspace of a normed space is finite dimensional, then it is closed.

Finally, we stated the important Riesz Lemma (R&Y 2.25), which is used in proving that the unit ball or sphere is compact precisely when the dimension is finite. This is a topological characterization of finite dimensionality of normed spaces (note the blend of topology and linear algebra!).

Read the rest of the second chapter in R&Y, starting with Example 2.7 and have a look at the exercises and their solutions. Note exercise 2.12 on page 50, demonstrating that the closure of an open ball in a normed space is the closed ball with the same center and radius (cf. a homework exercise in the first series: this is not true for arbitrary metric spaces).