GATE 2026 MA – Question 53
For a normed linear space X and its continuous dual X′, which statements hold?
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Correct answer: (B) If X is separable and X′ nonseparable, X cannot be reflexive.; (C) If X is reflexive, X′ is reflexive.; (D) If X′ is separable, X is separable.
Explanation
Dual separability implies separability of the underlying normed space, a consequence of Hahn–Banach, proving D. If X is reflexive, its dual is reflexive, proving C, and X″ is isomorphic to X. Were X separable and reflexive, X″ would be separable; applying the dual-separability theorem to X′ would make X′ separable. Thus a separable X with nonseparable dual cannot be reflexive, proving B and refuting A. **B,C,D**.