Functional Analysis MCQs

\[ \textbf{Difficult MCQs on Functional Analysis with Answers} \] \[ \textbf{Q1: What is the definition of a Banach space?} \] \[ \text{(A) } \text{A vector space equipped with a norm where every Cauchy sequence converges.} \] \[ \text{(B) } \text{A vector space equipped with a norm where every sequence is bounded.} \] \[ \text{(C) } \text{A vector space equipped with an inner product.} \] \[ \text{(D) } \text{A vector space equipped with a continuous linear functional.} \] \[ \textbf{Answer: (A) A vector space equipped with a norm where every Cauchy sequence converges.} \] \[ \textbf{Q2: Which of the following is a property of a Hilbert space?} \] \[ \text{(A) } \text{It is a complete inner product space.} \] \[ \text{(B) } \text{It is a finite-dimensional vector space.} \] \[ \text{(C) } \text{It must contain only continuous linear operators.} \] \[ \text{(D) } \text{It is a normed space with a bounded sequence.} \] \[ \textbf{Answer: (A) It is a complete inner product space.} \] \[ \textbf{Q3: In functional analysis, what is the dual space \(X^*\) of a normed vector space \(X\)?} \] \[ \text{(A) } \text{The set of continuous linear functionals on \(X\).} \] \[ \text{(B) } \text{The set of bounded operators on \(X\).} \] \[ \text{(C) } \text{The set of all continuous functions on \(X\).} \] \[ \text{(D) } \text{The set of all finite-dimensional subspaces of \(X\).} \] \[ \textbf{Answer: (A) The set of continuous linear functionals on \(X\).} \] \[ \textbf{Q4: Which of the following statements about bounded linear operators between Banach spaces is true?} \] \[ \text{(A) } \text{A bounded linear operator is continuous.} \] \[ \text{(B) } \text{A bounded linear operator is always invertible.} \] \[ \text{(C) } \text{A bounded linear operator is always surjective.} \] \[ \text{(D) } \text{A bounded linear operator is always injective.} \] \[ \textbf{Answer: (A) A bounded linear operator is continuous.} \] \[ \textbf{Q5: What is the spectral radius of an operator \(T\) in a Banach space?} \] \[ \text{(A) } \text{The largest eigenvalue of \(T\).} \] \[ \text{(B) } \text{The supremum of the absolute values of the eigenvalues of \(T\).} \] \[ \text{(C) } \text{The norm of the operator \(T\).} \] \[ \text{(D) } \text{The trace of the operator \(T\).} \] \[ \textbf{Answer: (B) The supremum of the absolute values of the eigenvalues of \(T\).} \] \[ \textbf{Q6: In the context of a normed vector space, what does the Hahn-Banach theorem assert?} \] \[ \text{(A) } \text{Any continuous linear functional can be extended to the entire space.} \] \[ \text{(B) } \text{Every bounded linear operator is continuous.} \] \[ \text{(C) } \text{A linear operator can be decomposed into a sum of two bounded operators.} \] \[ \text{(D) } \text{The dual of a Banach space is a Hilbert space.} \] \[ \textbf{Answer: (A) Any continuous linear functional can be extended to the entire space.} \] \[ \textbf{Q7: Which of the following is true about the Riesz representation theorem?} \] \[ \text{(A) } \text{It provides a one-to-one correspondence between bounded linear functionals and vectors in a Hilbert space.} \] \[ \text{(B) } \text{It applies only to finite-dimensional spaces.} \] \[ \text{(C) } \text{It applies only to continuous linear functionals on Banach spaces.} \] \[ \text{(D) } \text{It provides a decomposition of a functional into orthogonal components.} \] \[ \textbf{Answer: (A) It provides a one-to-one correspondence between bounded linear functionals and vectors in a Hilbert space.} \] \[ \textbf{Q8: What is the operator norm of a bounded linear operator \(T\) from a normed space \(X\) to \(Y\)?} \] \[ \text{(A) } \text{The supremum of \(\|T(x)\|\) for all \(x \in X\) with \(\|x\| = 1\).} \] \[ \text{(B) } \text{The infimum of \(\|T(x)\|\) for all \(x \in X\) with \(\|x\| = 1\).} \] \[ \text{(C) } \text{The supremum of \(\|T(x)\|\) for all \(x \in X\) with \(\|x\| \leq 1\).} \] \[ \text{(D) } \text{The infimum of \(\|T(x)\|\) for all \(x \in X\) with \(\|x\| \leq 1\).} \] \[ \textbf{Answer: (A) The supremum of \(\|T(x)\|\) for all \(x \in X\) with \(\|x\| = 1\).} \] \[ \textbf{Q9: In a Hilbert space, what is the result of the orthogonal projection of a vector \(x\) onto a subspace \(M\)?} \] \[ \text{(A) } \text{It is the closest point in \(M\) to \(x\).} \] \[ \text{(B) } \text{It is the point in \(M\) that is farthest from \(x\).} \] \[ \text{(C) } \text{It is the vector in \(M\) that has the same norm as \(x\).} \] \[ \text{(D) } \text{It is a vector orthogonal to \(M\).} \] \[ \textbf{Answer: (A) It is the closest point in \(M\) to \(x\).} \] \[ \textbf{Q10: Which of the following is true for the space \(L^p\) for \(1 \leq p < \infty\)?} \] \[ \text{(A) } \text{It is a Banach space.} \] \[ \text{(B) } \text{It is a Hilbert space.} \] \[ \text{(C) } \text{It is a finite-dimensional space.} \] \[ \text{(D) } \text{It is not a vector space.} \] \[ \textbf{Answer: (A) It is a Banach space.} \]