Prove that
l¹⊂
l² If: T :(l² ||.||)(l¹,||.||₂)(l^1,||.||₂) is a compact operator, show that: T:,||.||||.||₂ is also compact.

Find a bounded linear functional f on 3 l such that 3 e(f ) 3 = and || f|| = .3

Let Y A : X0 ⊆ X → be a closed operator where X and Y are Banach spaces. Define x x Ax , x X . A = + ∈ 0 Then show that the norm A ⋅ is complete.

Define 3 3 A : C → C by A z( , z , z ) iz( , e z , z ) 2 3

Check whether the identity map on an infinite dimensional space is compact.

Let X and Y be Banach spaces and :F X → Y be a linear map which is continuous and open. Will F always be closed? Will F be always surjective? Give reasons for your answer.

Let ]1,0[ X = C′ and ] Y = 11,0[C and let T : X → Y be the linear operator from X to Y given by ,f )f(T = ′ the derivative of f on ].1,0[ Show that T is not continuous.

Let H be a Hilbert space. For any subset A of ,H define . A ⊥ If ,H A ⊆ B ⊆ then show that: 

i)  B¹⊆ A

ii)  A ⊆ A ^⊥⊥

State conditions on A^⊥⊥ =A

Let X be a vector space. Let be two norms on .X When are these norms said to be equivalent? Justify your answer.

Let . X  = For  x=(x_1,x₂,x₃)

Let

Show that and are equivalent.

Let X be an inner product space with the inner product given by <, > . For ,X x ∈ define the function given by the non negative square root of < x, x > . Show that defines a norm on X and for all .X ,x y∈ Also show that for all ,X ,.

Let be a norm on a linear space .X if X ,x y∈ and then show that for all .0 s ≥ t,0 ≥ .

Define Eigen Spectrum of a bounded linear operator on a Banach space. Show that the eigen spectrum of the operator T on  l² given by ) (T ,......... ) =  is empty .

Let T be a linear map defined on X by

Show that T is a bounded linear map such that T = .1

Let ]1,0[C X = with Sup norm defined by f = Sup{| )x(f |}. ]1

Let } u{ n be the sequence in  l² with 1 in the th n place and zeroes else where prove that the set } u{ n is an orthonormal basis for l ²

Let A be an operator on a Hilbert space .H Show if Ax A x ∗ = for every ,H x ∈ then A is normal. Is it converse true? Justify

Show that Q defined on by dt)t(xt is a bounded linear functional. Calculate .

If H is a Hilbert space and SCH, show that . S S ⊥ ⊥⊥⊥ = When S is the same as ? S ⊥⊥ Justify.

State the principle of uniform boundedness. Use it to show that a set E in a normed space X is bounded if )E(f is bounded in K for every f ∈ X′.

Are Hahn-Banach extensions always unique? Justify.