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