GATE 2022 MA – Question 43

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

X,Y,Z are Banach spaces, T:X→Y linear, and S:Y→Z bounded linear and injective. If S∘T is bounded, which must hold?

T is surjective.
T is bounded but not continuous.
T is bounded.
T is not bounded.

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Show answer and explanation

Correct answer: (C) T is bounded.

Explanation

Prove T has a closed graph. If x_n→x and Tx_n→y, continuity of S and ST gives Sy=lim STx_n=STx. Injectivity of S yields y=Tx. The closed-graph theorem for Banach spaces therefore gives boundedness of T. **C**.

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