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 (X, {\mathcal X}, \mu) be a \sigma-finite measure space, and let \nu: {\mathcal X} \to {\Bbb R} be a signed \sigma-finite measure.  Show that the following are equivalent:

  1. -\mu \leq \nu \leq \mu.
  2. |\nu| \leq \mu.
  3. \nu \ll \mu and |\frac{d\nu}{d\mu}| \leq 1 \mu-a.e.. \diamond

Question 2. Let ({\Bbb R}, {\mathcal B}) be the real line with the Borel \sigma-algebra, and let \mu: {\mathcal B} \to [0,+\infty) be a \sigma-finite unsigned measure.  Show that it is not possible for counting measure \# on the real line (restricted to {\mathcal B}) to be absolutely continuous with respect to \mu, i.e. \# \not \ll \mu.  \diamond

Question 3. Let 1 \leq p < \infty, let \ell^p({\Bbb N}) be the Banach space of p^{th} power summable real sequences (a_n)_{n \in {\Bbb N}} with the usual \ell^p norm.  For each natural number n, let e_n be the element of \ell^p({\Bbb N}) consisting of the sequence which equals 1 at the n^{th} entry and 0 elsewhere, thus (e_n)_m = 1 when n=m and (e_n)_m =0 otherwise. Let a_1, a_2, a_3, \ldots be a sequence in a Banach space X.

  1. Show that there is at most one continuous linear transformation T: \ell^p({\Bbb N}) \to X with the property that T e_n = a_n for all natural numbers n.
  2. If p=1, show that there exists a continuous linear transformation T: \ell^1({\Bbb N}) \to X with T e_n = a_n for all natural numbers n if and only if the sequence a_1, a_2, \ldots is bounded. \diamond
  3. If p=\infty, show that the uniqueness claim 1. can fail. \diamond

Question 4. Let W be a vector space, let A be an index set, and for every \alpha \in A, let V_\alpha be a subspace of W which is equipped with a norm \| \|_{V_\alpha}.  Suppose that for each \alpha, the normed vector space (V_\alpha, \| \|_{V_\alpha}) is a Banach space.  Assume also the following compatibility condition: if \alpha, \beta \in A and x_n is a sequence in V_\alpha \cap V_\beta which converges in V_\alpha norm to x and in V_\beta norm to y, then x is necessarily equal to y.  Show that the space

V := \{ x \in \bigcap_{\alpha \in A} V_\alpha: \sum_{\alpha \in A} \| x\|_{V_\alpha} < \infty \},

equipped with the norm

\| x \|_V = \sum_{\alpha \in A} \|x\|_{V_\alpha},

is also a Banach space. \diamond

Question 5. Let H be a Hilbert space, and let

V_1 \subset V_2 \subset V_3 \subset \ldots \subset H

be an increasing sequence of closed subspaces of H.  Let V := \overline{\bigcup_{n=1}^\infty V_n} 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 \pi_{V_n}(x) converges in norm to \pi_V(x), where \pi_V(x) is the orthogonal projection of x to V, and similarly for \pi_{V_n}(x). \diamond

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 \{0\}).  Let \pi_V be the orthogonal projection onto V.

  1. Show that \pi_V has operator norm 1.
  2. Show that \pi_V is self-adjoint, i.e. \pi_V^* = \pi_V.
  3. Show that \pi_V is idempotent, i.e. \pi_V^2 = \pi_V.
  4. Conversely, if P: H \to H is a self-adjoint idempotent continuous linear transformation of operator norm 1, show that P = \pi_V for some non-trivial subspace of H. \diamond

[Update, Feb 2: Question 4 corrected.]

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