Let be Banach (more generally Fréchet) spaces and let be continuous linear maps. Suppose that is surjective and is compact. Prove that has closed image and .
Let be an infinite dimensional normed space. There is no non-trivial translation invariant Borel measure on which is finite on open balls.
Let and be subintervals of and call them non-overlapping if they intersect only possibly in their endpoints. Construct a continuous curve with a Hilbert space, such that if and are any non-overlapping subintervals of , then and are orthogonal.
Harder: show that this can be carried out in any infinite-dimensional Hilbert space.
Prove the following remarkable properties of and
- Given any normed space and a subspace , any continuous linear map may be extended to all of preserving its norm.
- Every separable Banach space is a quotient of