GATE 2023 MA – Question 37

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

On real ℓ² with usual norm, S(x₁,x₂,…)=(0,x₁,x₂,…) and T(x₁,x₂,…)=(0,x₂,0,x₄,…). Which holds?

S compact, T not compact
S not compact, T compact
Both compact
Neither compact

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Show answer and explanation

Correct answer: (D) Neither compact

Explanation

For the standard orthonormal basis e_n, Se_n=e_(n+1) stays pairwise √2 apart. Also Te_(2n)=e_(2n) stays √2 apart. Both maps send a bounded sequence to one without a convergent subsequence, so neither is compact. **D**.

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