Reference no: EM132914308

Mathematical Analysis

Question 1. Show that all hilbert space H is reflexive.

Question 2. Let be H a hilbert space and M a subspace of H. Show that (M^⊥)^⊥ = M

Question 3. a) Let be a space C(S¹) with S¹ = {z ∈ C : |z| = 1}. Show that finite sums Σ^l_k=-m a_kz^k, with a^k ∈ C is dense in C(S¹). Hint: Use an appropiate version of Stone-Weierstrass theorem.

b) Show family {sin(nx), cos(nx)}n∈N is an hilbertian base in L²(0, 2Π).

Question 4. Let be H hilbert space and V ⊂ H closed subspace. Show that if φ ∈ V∗ then there exists a single continuous linear extension of φ defined in H that preserve the norm.

Question 5. Let be H separable hilbert space, with {e_n}n∈N base. T is an Hilbert-Smith operator if T : H → H is linear and continue and Σ_n∈N |T (e_n)|² is finite. Show that T is a compact operator.

Question 6. Let be 1 ≤ p ≤ ∞ and {λ_n} bounded. We define T : L^p(N) → L^p(N) defined by T ({x_n}) = {λ_nx_n}n∈N. Show that T is compact if and only if λ_n → 0 when n → ∞.

Question 7. Consider a sequence of positive numbers {λ_n} → ∞. We define the space V of the sequences {x_n}n∈N such that

Σλ_n|xn|² < ∞
n∈N

and V is such that < {x_n}, {y_n} >= Σ_n∈N λ_nx_ny_n
a) Show that V is Hilbert.
b) Show that the injection i : V → L²(N) is compact.

Question 8. Let be E and F Banach spaces, and T : E → F compact, and T (E) closed.
a) Show that T (E) has finite dimension.
b) Show that if also dim(Ker(T )) < ∞ then E is finite dimension.