Question 1: Show that B(X,Y) = {T: T: X → Y; T is bounded and linear operator} forms a normed spaces with the norm ||T|| = sup { ||Tx|| / ||x|| : x ∈ X, x ≠ 0 }.
Question 2: Define Dual space of a normed space and show that the dual of l! is l∞.
Question 3: Show with the help of example that C[0,1] does not satisfy the parallelogram law with the norm ||x|| = sup |x(t)|.
Question 4: Let X and Y are Banach spaces with the norms ||!!!! and ||!!!!, respectively. Then
(a) Show that X × Y is a normed space with the norm ||x||* = max{||x1||1, ||x2||2}.
(b) Show that X × Y forms a Banach space with the norm ||x||* = ||x1||1 + ||x2||2.
Question 5: If M is closed subspace of a Hilbert space X, then show that X = M ⊕ M⊥.
Question 6: Let X and Y are normed spaces and T : X → Y be an operator. Then show that the following conditions are equivalent.
(a) ||T|| = sup { ||Tx|| / ||x|| : x ∈ X, x ≠ 0 }.
(b) ||T|| = sup { ||Tx|| : x ∈ X, ||x|| = 1 }.
Question 7: Let X and Y are normed spaces and T : X → Y be a linear operator. Then show that the following conditions are equivalent.
(a) T is continuous.
(b) T is continuous at origin.
(c) T is bounded.
Question 8: Give example (at least one) of bounded operator, unbounded operator, adjoint operator, self adjoint operator, unitary operator and Normal operator.