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.)
Let us define a topology on the real line in the following way: By definition, a set is −open if an only if for each there is a compact subset such that
where, for a Lebesgue-measurable set , we denote by its Lebesgue measure.
Verify that the definition actually gives a topology.
Show that any −open set is Lebesgue measurable. (Recall a set is Lebesgue measurable if it can be covered by an open set such that the outer measure of is arbitrarily small. Use the infinite version of the Vitali Covering Theorem.)
Is every −open set a Borel set? (Take the complement of a measure zero subet that is not Borel measurable; note that Lebesgue measure is inner regular)
Is the real line connected in the −topology? (Yes. Note that for fixed the function is continuous, and if we can disconnect in the -topology, then takes values close to 0 and close to 1. Thus we can find such that . We can repeat the same argument with some , since the average of in is close to if is sufficiently small. Taking , we see that either doesn’t exist or .)
Show that cannot be the set of points of continuity of a real-valued function . (Such set must be , i.e. a countable intersection of open subsets, but is not by Baire Category.)
Use double integral to compute . (The key identity is . To justify the change of order of integration, we use Fubini-Torelli Theorem. See this answer for detail.)
By Holder’s inequality, we have . Show that it is of first category. (Take for and 0 otherwise. Then for any , we have in by Holder. Thus where . Each is closed and has empty interior, since we can construct a function such that and for all .)
Show that the analogue of invariance of domain for infinite-dimensional Banach space is false, i.e. find a Banach space and a continuous injective self-map such that the image is not open. (Take the right shift .)
Show that if is absolutely continuous, then (a) it preserves any measure zero susbets. (This is almost immediate by definition of absolute continuity, the only trick being that given a finite covering by intervals of total length less than , we can further refine it so that is contained in an interval of length close to by continuity.) (b) it preserves measurable subsets. (Any Lebesgue measurable subset of can be written as the disjoint union of a Borel measurable subset of the same measure and a measure zero subset by outer regularity. The image of Borel subset under continuous functions, so called analytic set, is Lebesgue measurable. This is an entry-point of descriptive set theory, for a proof see this answer.)