Real Analysis II.

Instructor: Dmitry Ryabogin

Assignment XI.

1. Problem 1. Let 0 ≤ t < 1 and let x_{t} = (1, t, t^{2}, ..., t^{k}, ...,) ∈ l^{2} = H. Compute the distance between x_{t_{0}} and span{x_{t_{1}}, ..., x_{t_{m}}}, where {t_{j}}_{j=0}^{m} are different from each other. Hint:

G(x_{t_{1}}, ..., x_{t_{m}}) = (t_{1}...t_{m})^{−1} det ((t_{j}^{−1} − t_{i})^{−1})_{1≤i,j≤m}.

2. Problem 2. Let P, Q be projections on closed subspaces M, N of H.

a) Prove that Im(P Q) = {y ∈ H : P Qy = y} = M ∩ N, provided P Q is a projection.

b) Prove that P Q is a projection iff P Q = QP.

c) Prove that P + Q − P Q is a projection, provided P Q is a projection.

d) Prove that Im(P + Q − P Q) = {y ∈ H : (P + Q − P Q)y = y} = M + N.

3. Problem 3.

Let M be a closed subspace of H. Prove that for every x_{0} ∈ H, we have

min {∥x − x_{0}∥; x ∈ M} = max {(x_{0}, y); ∥y∥ = 1, y ∈ M^{⊥}}.

4. Problem 4. In this exercise a ”projection” is a linear bounded operator P : H → H, satisfying P^{2} = P.

a) Prove that ImP, KerP are closed subspaces of H and that H = ImP + KerP.

b) Prove the converse: If H = M +N, M ∩N = {0}, where M, N are closed subspaces of H, then there exists a unique projection P with M = ImP, N = KerP.

c) Prove that P is a projection iff I − P is a projection.

d) Let a > 0. Give an example of a projection P satisfying ∥P ∥ > a.

e) Prove that a projection P is orthogonal, (∥P x∥ ≤ ∥x∥), iff (P x, y) = (x, P y) for all x, y ∈ H.

5. Problem 5. a) Let H = C^{n}. Assume that the matrix of a linear operator T (with respect to a standard basis) is given as (a_{ij})_{1≤i, j≤n}. Prove that ∥T ∥^{2} ≤ ∑_{i,j=1}^{n} |a_{i j}|^{2}.

b) Let (a_{ij})_{i,j=1}^{∞} be an infinite Hilbert-Schmidt matrix, ∑_{i,j=1}^{∞} |a_{ij}|^{2} < ∞. Prove that for every i and for every x = (x_{1}, x_{2}, ...) ∈ l^{2}, y_{i} = ∑_{j=1}^{∞} a_{ij}x_{j} is convergent and

y = (y_{1}, y_{2}, ...) ∈ l^{2}. Let T x = y. Prove that T is bounded and ∥T ∥^{2} ≤ ∑_{i,j=1}^{∞} |a_{ij}|^{2}.