Functional Analysis-Ii

Hilbert adjoint operator T* satisfies which of the following conditions ?
(a) <Tx, y> = <x, T*^-1y>
(b) <T*x, y> = <x, T*y>
(c) <Tx, y> = <x, T*y>
(d) <Tx, y> = 0

If T is an operator on Hilbert space H with <Tx, x> = 0; ∀ x then :
(a) T = 0
(b) T ≠ 0
(c) T is not defined
(d) None of the above

A partially ordered set A, which is finite has at least how many maximal element ?
(a) 0
(b) 1
(c) 2
(d) ∞

Real valued functional p on a vector space X is subadditive if :
(a) p(x + y) = p(x) + p(y)
(b) p(x + y) > p(x) + p(y)
(c) p(x + y) ≤ p(x) + p(y)
(d) p(x + y) < p(x) + p(y)

If T is represented by a Matrix then the adjoint operator T* is represented by :
(a) Inverse of Matrix
(b) Transport of Matrix
(c) Determinant
(d) None of the above

If a normed space X is reflexive then it is:
(a) Vector space
(b) Banach Space
(c) Metric Space
(d) Topological Space

A subset M of a metric space X is said to be nowhere dense in X if its closure M̅ is :
(a) no interior points
(b) many interior points
(c) limit points
(d) None of the above

A sequence (x_n) in a normed space X is said to be strongly convergent if ∃ x ∈ X such that :
(a) ||x_n - x|| = 0
(b) ||x_n - x|| ≥ 0
(c) lim_n→∞ ||x_n - x|| = 0
(d) lim_n→∞ ||x_n - x|| = ∞

If T ∈ B(X, Y), where X is Banach space and Y a normed space. <T_n> is strongly operator convergent with limit T; then :
(a) T ∈ B(X, Y)
(b) T ∈ B(X, X)
(c) T ∈ B(Y, Y)
(d) None of the above

If (X, d) be a metric space. A mapping T : X → X is called contraction ion X if ∃ positive real number α < 1 such that ∀ x, y ∈ X :
(a) d(T_x, T_y) ≥ α d(x, y)
(b) d(T_x, T_y) > α d(x, y)
(c) d(T_x, T_y) < α d(x, y)
(d) d(T_x, T_y) ≤ α d(x, y)

If the operators U : H → H and V : H → H be unitary, where H is a Hilbert space then prove that :

U is Normal

A bounded linear operator T on a complex Hilbert space H is unitary iff T is isometric and subjective.

Define Partially ordered set and normed linear space. Write statement of Zorn's Lamma.

If f(x) = f(y) for every bounded linear functional f on a normed space X, then show that x = y.

Define Canonical mapping and Reflexivity.

Prove that every Hilbert space is Reflexive.

If <x_n> be a sequence in a Normed space X, if dim X < ∞ then prove that weak convergence implies strong convergence.

Show that in a Normed space X, x_n —^w→ x if and only if for every element f of a total subset M ⊂ X' we have f(x_n) → f(x).

State and prove open mapping theorem.

Define convergence of sequences of operators and functionals.

Prove that Hilbert adjoint operator T* of T exists, it is unique and is a bounded linear operator with norm .

If T : H → H be a bounded linear operator on a Hilbert space H, then prove that :

If T is self adjoint <Tx, x> is real ∀ x ∈ H.

If H is complex and <Tx, x> is real, ∀ x ∈ H, operator T is self adjoint.

State and prove Hahn Banach theorem for Real Linear Spaces.

State and prove Hahn-Banach theorem for complex Vector spaces.

Explain the relation between the adjoint operator T* and the Hilbert adjoint operator T*.

If the Dual Space X' of a Normed space X is separable then prove that X itself is separable.

State and prove Baire's category theorem.

State and prove Uniform Boundedness theorem.

State and prove closed Graph theorem.

State and prove contraction theorem.