Reference no: EM132926024

Functional Analysis Assignment

Question 1. (a) Let H be a Hilbert space and M₁, M₂ two vector subspaces of H. Prove that
M₁^⊥ ∩ M₂^⊥ = (M₁ = (M₁ + M₂)^⊥.
Recall that M₁ +M₂ := {m₁ +m₂ |m₁ ∈ M₁, m₂ ∈ M₂}.

(b) Suppose that H is a Hilbert space and P ∈ £(H) such that P² = P and P = P*. Show that P is an orthogonal projection.

(c) Let α = (α_k)k≥1 be a bounded sequence in C. Let 1 < p < ∞ and let T: l^P → l^P be defined by T_x := (α_kx_k)_k≥1 for all complex sequences x = (x_k)_k≥1 ∈ l^P.

(i) Determine the spectrum of T.

(ii) Show that T is compact if and only if α ∈ c₀.

Question 2. (a) Let (a, b) ⊆ R be a bounded open interval. Let

S:= {u:(a, b) --> R ||u||_L < ∞},

where

||u||_L := sup_t,s∈(a,b) |u(t) - u(s)|/|t-s|

(i) Show that every u ∈ S is bounded.
(ii) Decide whether or not ||.||_L defines a norm on the set S.
(iii) Define E:= {u : (a, b) -> R | ||u|| < ∞), where ||u||:= ||u||_∞ + ||u||_L show that (E, ||.||) is a complete normed vector space. You may use that B ((a, b)) is complete with respect to the supremum norm.

(b) Let E be a Banach space and (T_α)_α∈A a family of linear operators in £(E). Assume that

sup_α∈A l<f , T_αx>) | < ∞

for all f ∈ E' and all x ∈ E. Show that sup_α∈A||T||_£(E)< ∞

Question 3. (a) Suppose that E and F are Banach spaces with duals E' and F'. Let T: E -> F and S: F' -> E' be linear operators such that (S f,x) = (f,Tx) for all x ∈ E and all f ∈ F'. Show that T ∈ L(E,F).

(b) Define the map T: l¹ -> l¹ by

T_x =T(x₁,x₂,.. ):= (0,x₁,x₂/2,x₃/3, x₄/4, ....)

for all x = (x_k)_k≥1 ∈ l¹, that is, Tx = (Y_k)_k≥1 with y = 0 and y_k+1 = X_k/k if k ≥ 1.

(i) Show that T ∈ L(l¹) and determine the operator norm ||T"||£(^l1) for all n ∈ N.

(ii) Find the spectrum of T.