Give an example of a positive operator on .

Give an example of a compact linear map on l².

Let X be a Banach space and Y be a closed subspace of X. Let π: X → X/Y be canonical quotient map. Show that is open.

Define the spectral radius of a bounded linear operator A ∈ BL(X). Find the spectral radius of A in BL, where A is given by the matrix

with respect to the standard basis of .

Let {A_n} be a sequence of unitary operators in BL(H). Prove that if , then A is unitary.

Define a Hilbert-Schmidt operator on a Hilbert space H and give an example. Is every Hilbert-sehmidt operator a compact operator? Justify your answer.

Let X,Y be normed spaces and suppose BL(X,Y) and CL(X,Y) denote, respectively, the space of bounded linear operators from X to Y and the space of compact linear operators from X to Y. Show that CL(X,Y) is linear subspace of BL(X,Y). Also, Show that if Y is a Banach space, then CL(X,Y) is a closed subspace of BL(X,Y).

Let X be a normed space and Y be proper subspace of X. Show that the interior Yº of Y is empty.

Give one example of each of the following. Also justify your choice of example.

i) A self-adjoint operator on .

ii) A normal operator on a Hilbert space which is not unitary.

Let X = C₀₀ with  Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example.

Let A be a normal operator on a Hilbert space X. Show that σ(A) ⊂ σ_a(A) where σ_a (A) denotes the approximate eigen spectrum of A and σ(A) denotes the spectrum of A.

Given an example of an Hilbert space H and an operator A on Η such that σ_e(A)is empty. Justify your choice of example.

Let H=R³ and F be the set of all x = (x₁, x₂, x₃) in H such that x₁ = 0. Find F¹. Verify that every x ∈ H can be expressed as x = y + z where y ∈ Fand z ∈ F¹.

Let X be an inner product space and x, y ∈ X. Prove that x | y if and only if

Let f: C[0,1]→ be given by f (x) = x(1)∀x ∈ C[0,1]. Show that f is continuous w.r.t the supnorm and f is not continuous w.r.t the p-norm.

Which of the following maps are open? Give reasons for your answer.

i)

ii)

Read the proof of the closed graph theorem carefully and explain where and how we have used the following facts in the proof.

i) X is a Banach space.

ii) Y is a Banach space.

iii) F is a closed map.

iv) Which property of continuity is being established to conclude that F is continuous.

Let X be a Banach space, Y be a normed linear space and be a subset of B(X, Y). If is not uniformly bounded, then there exists a dense subset D of X such that for every is not bounded in Y.

When is a normed linear space called separable? Show that a normed linear space is separable if its dual is separable [You should state all the proposition or theorems or corollaries used for proving the theorem]. Is the converse true? Give justification for your answer. [Whenever an example is given, you should justify that the example satisfies the requirements.]

Consider the space C¹[0,1] of all C¹ functions on [0,1] endowed with the uniform norm induced from the space C[0,1], and consider the differential operator defined by Df = f'. Prove that D is linear, with closed graph, but not continuous. Can we conclude from here that C¹[0, 1] is not a Banach space? Justify your answer.