Real Analysis II.

Instructor: Dmitry Ryabogin

Assignment XII.

1. Problem 1. Let e_{n} = (0, ..., 0, 1, 0, ..., 0, ...), (1 stands on nth place), n = 1, 2, ..., be a sequence of vectors of l^{2}. In the case H = l^{2} give a direct proof of the Riesz theorem by showing that for any linear bounded functional on l^{2} we have

∑_{n=1}^{∞} |f(e_{n})|^{2} < ∞.  (1)

Hint: Assume that (1) is not true, then there are disjoint subsets E_{k} of natural numbers, k = 1, 2, ..., so that

∑_{n∈E_{k}} |f(e_{n})|^{2} > 1.

Define x_{n} so that x_{n} = c_{k}f(e_{n}) for n ∈ E_{k}. For suitably chosen c_{k},

∑_{n=1}^{∞} x_{n}f(e_{n}) = ∞,

although

∑_{n=1}^{∞} |x_{n}|^{2} < ∞.

2. Problem 2. Let {x_{n}}, n = 1, 2, 3, ... be a linearly independent system (set) of vectors on H.

a) Show that the following construction yields an orthonormal system u_{n} such that {x_{1}, ...x_{N}} and {u_{1}, ..., u_{N}} have the same span for all N. Hint: Put u_{1} = x_{1}/∥x_{1}∥. Having u_{1}, ..., u_{n−1} define

v_{n} = x_{n} − ∑_{k=1}^{n−1}(x_{n}, u_{k})u_{k}, u_{n} = v_{n}/∥v_{n}∥.

Note that this leads to a proof of the existence of a maximal orthonormal system in separable Hilbert spaces.

b) Let x_{n} = x^{n−1}, n = 1, 2, 3, ... be a linearly independent system of vectors in L^{2}(0, 1). What are u_{n}?

c) Let x_{1} = (1, 0, ..., 0, ...), x_{2} = (1, 1/2, 0, ..., 0, ...), x_{3} = (1, 0, 1/4, 0, ..., 0, ...),..., x_{n} = (1, 0, ..., 0, 1/2^{n−1}, 0, ..., 0, ...),... be a system of vectors in l^{2}. Are these vectors independent? Is this system orthonormal? If not, what are u_{n}? Is it true that the closure of the linear span of (x_{j})_{j=1}^{∞} is l^{2}? Is (x_{j})_{j=1}^{∞} a basis?

d) Prove that in L^{2}([a, b]) there are orthonormal systems consisting of polynomials, piecewise linear functions, trigonometric functions.

3. Problem 3. a) If A ⊂ [0, 2π], and A is measurable, prove that

lim_{n→∞} ∫_{A} cos(nx)dx = lim_{n→∞} ∫_{A} sin(nx)dx = 0.

Hint: 1_{A} ∈ L^{2}([0, 2π]).

b) Let n_{1} < n_{2} < n_{3} < ... be positive integers, and let E be the set of all x ∈ [0, 2π] at which {sin(n_{k}x)} converges. Prove that μ(E) = 0. Hint: 2 sin^{2} α = 1 − cos(2α), so sin(n_{k}x) → ±1/\sqrt{2} a.e. on E.