Functional Analysis

Q1: Let X be a Banach space and T: X → X be a compact linear operator. Which of the following is TRUE about the spectrum σ(T)?

σ(T) is finite σ(T) is countable with 0 as the only possible accumulation point σ(T) is uncountable σ(T) is the entire complex plane

Q2: In a Hilbert space H, which of the following is TRUE about an orthonormal basis?

Every orthonormal basis has the same cardinality Every orthonormal basis is countable Every orthonormal basis is finite There exists at most one orthonormal basis

Q3: Let X and Y be normed spaces and T: X → Y be a bounded linear operator. Which of the following is TRUE?

If T is injective, then T is an isometry If T is surjective, then T is an open mapping The graph of T is closed if X is Banach T is continuous iff its graph is closed

Q4: Which of the following is NOT a Hilbert space?

L²[0,1] l² (square summable sequences) C[0,1] with sup norm ℓ² with standard inner product

Q5: Let X be a normed space. Which of the following is TRUE about the dual space X'?

X' is always reflexive X' is always a Banach space X' is always separable X' is always finite-dimensional

Q6: In a Banach space X, which of the following is TRUE about weakly convergent sequences?

Every weakly convergent sequence is norm convergent Every weakly convergent sequence is bounded Weakly convergent sequences are always Cauchy Weakly convergent sequences have unique limits only in reflexive spaces

Q7: Let H be a Hilbert space and M be a closed subspace. Which of the following is TRUE?

H = M ⊕ M^⊥ H = M + M^⊥ H is isomorphic to M × M^⊥ All of the above

Q8: Which of the following operators on a Hilbert space is NOT necessarily bounded?

Compact operators Adjoint operators Unitary operators Projection operators onto closed subspaces

Q9: What is the dual space of L^p[0,1] for 1 < p < ∞?

L^q[0,1] where 1/p + 1/q = 1 L^∞[0,1] L¹[0,1] C[0,1]

Q10: Let T: X → Y be a bounded linear operator between Banach spaces. Which of the following is TRUE about the adjoint operator T': Y' → X'?

T' is compact if T is compact T' is compact iff T is compact T' is always compact T' is never compact

Q11: In a Banach algebra A with identity, which of the following is TRUE about invertible elements?

The set of invertible elements is open The set of invertible elements is closed The set of invertible elements is compact The set of invertible elements is connected

Q12: Let X be a reflexive Banach space. Which of the following is TRUE?

Every bounded sequence has a weakly convergent subsequence Every weakly convergent sequence is strongly convergent X is separable X has the Schur property

Q13: Which of the following is TRUE about the spectrum of a bounded linear operator?

The spectrum is always non-empty The spectrum is always finite The spectrum is always countable The spectrum is always compact

Q14: In a normed space X, which of the following is TRUE about the closed unit ball?

It is compact iff X is finite-dimensional It is always compact It is compact iff X is reflexive It is never compact

Q15: Let T: X → Y be a bounded linear operator. Which of the following is TRUE about the kernel and range of T?

Ker(T) is closed and Im(T) is always closed Ker(T) is closed but Im(T) may not be closed Ker(T) may not be closed but Im(T) is closed Both Ker(T) and Im(T) are always closed