Below the fold, I am giving some sample questions for the 245B midterm next week. These are drawn from my previous 245A and 245B exams (with some modifications), and the solutions can be found by searching my previous web pages for those courses. (The homework assignments are, of course, another good source of practice problems.) Note that the actual midterm questions are likely to be somewhat shorter than the ones provided here (this is particularly the case for those questions with multiple parts). More info on the midterm can be found at the class web page, of course.
(These questions are of course primarily intended for my students than for my regular blog readers; but anyone is welcome to comment if they wish.)
Question 1. Let be a -finite measure space, and let be a signed -finite measure. Show that the following are equivalent:
- and -a.e..
Question 2. Let be the real line with the Borel -algebra, and let be a -finite unsigned measure. Show that it is not possible for counting measure on the real line (restricted to ) to be absolutely continuous with respect to , i.e. .
Question 3. Let , let be the Banach space of power summable real sequences with the usual norm. For each natural number n, let be the element of consisting of the sequence which equals 1 at the entry and 0 elsewhere, thus when and otherwise. Let be a sequence in a Banach space X.
- Show that there is at most one continuous linear transformation with the property that for all natural numbers n.
- If , show that there exists a continuous linear transformation with for all natural numbers n if and only if the sequence is bounded.
- If , show that the uniqueness claim 1. can fail.
Question 4. Let W be a vector space, let A be an index set, and for every , let be a subspace of W which is equipped with a norm . Suppose that for each , the normed vector space is a Banach space. Assume also the following compatibility condition: if and is a sequence in which converges in norm to x and in norm to y, then x is necessarily equal to y. Show that the space
equipped with the norm
is also a Banach space.
Question 5. Let H be a Hilbert space, and let
be an increasing sequence of closed subspaces of H. Let be the closure of the union of these subspaces; this is another closed subspace of H. Show that for any x in H, the sequence converges in norm to , where is the orthogonal projection of x to V, and similarly for .
Question 6. Let H be a Hilbert space, and let V be a closed subspace of H which is non-trivial (i.e. not equal to ). Let be the orthogonal projection onto V.
- Show that has operator norm 1.
- Show that is self-adjoint, i.e. .
- Show that is idempotent, i.e. .
- Conversely, if is a self-adjoint idempotent continuous linear transformation of operator norm 1, show that for some non-trivial subspace of H.
[Update, Feb 2: Question 4 corrected.]