GATE 2023 MA – Question 38

Functional Analysis · Bounded Linear Operators and Duality · 2 marks · Multiple choice

Let c₀₀ be finitely supported real sequences with supremum norm. Define F(x)_n=(n+1/n)x_n and G(x)_n=x_n/(n+1/n). Which holds?

Show answer and explanation

Correct answer: (B) F not continuous, G continuous

Explanation

‖e_n‖∞=1 but ‖Fe_n‖∞=n+1/n→∞, so F is unbounded and discontinuous. Since n+1/n≥2, ‖Gx‖∞≤‖x‖∞/2, giving a bounded continuous operator. **B**.

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