Prove proposition 3.2 assuming that X is finite-dimensional as follows: Let B be the unit ball in X, that is,
B = {x: ||x|| < 1}
The boundary of B is the unit sphere S = {x: ||x|| < 1}. Show that
1. f (S) is a compact subset of Y which does not contain 0Y.
2. There exists an open ball T ⊆ (f (S))c containing 0Y.
3. T ⊆ f(B).
4. f is open.
We now give three applications of proposition 3.2. The first formalizes a claim made in chapter 1, namely that the geometry of all finite-dimensional spaces is the same. There is essentially only one finite-dimensional normed linear space, and ℜn is a suitable manifestation of this space. The second (exercise 3.36) shows that a linear homeomorphism is bounded from below as well as above. The third application (exercise 3.37) shows that for linear maps, continuity is equivalent to having a closed graph (exercise 2.70). We will use proposition 3.2 again in section 3.9 to prove the separating hyperplane theorem.