This is the third post on qualifying exam preparation and it will be on questions in real analysis, measure theory, and some functional analysis.
- Let be the Hilbert space of real-valued integrable functions with inner product. Consider the linear operator given by .
Show that this map is continuous and invertible (with continuous inverse), and show that is self-adjoint.
Show that has no (non-zero) eigenvectors.
Fix any . Find a sequence such that for all and .
(This question is essentially definition-checking; For (c) let be supported on a interval of length containing .)
Given , exhibit an open subset of containing every rational number and having Lebesgue measure less than . (Enumerate the rationals and take the union of intervals of length around them.)
Show that the closed unit ball in is not compact. (Let if and otherwise. It is not Cauchy so cannot converge to anything.)