Due November 23
Let X be a metric space, the set of its nonempty compact subsets and the Hausdorff metric.
- 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 with the inner product
Apply the Gram-Schmidt process to the sequence to obtain the orthonormal polynomials , such that each is of degree n.
These are the first Legendre polynomials.
If are norms induced by inner products in , then they are equivalent: there exist constants such that
for all . (Hint: use the Gram-Schmidt process to construct orthonrmal bases for each inner product.)
Let X be an inner product space and its completion. Then is a Hilbert space.