Real Analysis.

Instructor: Dmitry Ryabogin

Assignment IX.

1. Problem 1.

a) Show that the vectors (x_{j})_{j=1}^{n} in H are linearly independent, provided they are pairwise orthogonal.

b) Prove that x = 0 iff (x, y) = 0 for all y ∈ H.

2. Problem 2.

Let H be a space of complex polynomials of degree ≤ n.

a) Show that the dimension of H is n + 1.

b) Let λ_{0} < λ_{1} < ... < λ_{n} be constants. Define

(P, Q) = ∑_{j=0}^{n} P (λ_{j})\overline{Q(λ_{j})}.

Prove that this is a scalar product, and prove that H is a Hilbert space with respect to it.

3. Problem 3. Matrices of Hilbert-Schmidt. Let M be a collection of all infinite matrices over complex numbers, a = ((a(i, j))_{1≤i,j<∞}, such that a(i, j) = 0 for n(a) = n < max{i, j}, (all terms a(i, j) are zero except for finitely many terms).

a) Prove that M is a linear space with respect to usual operations, and that one can define a multiplication in M. Define a trace of a ∈ M as follows

tr a = ∑_{i} a(i, i).

b) Let α, β be complex numbers, and let a, b ∈ M. Prove that tr ab = tr ba; tr a^{∗}a ≥ 0; tr (αa + βb) = αtr a + βtr b.

Here a^{∗} = (\overline{(a(j, i)})_{1≤i,j<∞}. For a, b ∈ M define

(a, b) = tr ab^{∗}.

c) Prove that this is a scalar product, and that

(a, b) = ∑_{i j} a(i, j)\overline{b(i, j)}.

d*) Show that, if we complete M with respect to this scalar product, we will get a collection of infinite matrices for which

∑_{i j} |a(i, j)|^{2} < ∞.

4. Problem 4*. Prove that the algebraic dimension of l^{2} is 2^{ℵ_{0}}. Hint: To get a lower bound look at a collection (x_{t})_{0≤t<1}, x_{t} = (1, t, t^{2}, t^{3}, ..., t^{k}, ...), and use the determinant of Vandermonde.

5. Problem 5. Give a precise formula for the projection P_{C}(x) := dist(x, C). a) C = B(z, r), b)

C = {x ∈ l^{2} : x_{i_{k}} = λ_{k}, k = 1, 2, 3, ...}, (λ_{k})_{k=1}^{∞} ∈ l^{2},

where i_{k} is an increasing sequence of of natural numbers.

c) C is a collection of all real non-negative (up to a set of measure zero) functions in L^{2}([0, 1]).