IGNOU MMT 6 2026 - Read the proof of the closed graph theorem carefully and explain where

Question

Read the proof of the closed graph theorem carefully and explain where and how we have used the following facts in the proof.
i) X is a Banach space.

ii) Y is a Banach space.
iii) F is a closed map.
iv) Which property of continuity is being established to conclude that F is continuous.

09 Jan 2026

Answer :

Word Count : 394

In the proof of the closed graph theorem, we consider a linear operator (F: X \to Y) between Banach spaces (X) and (Y), and assume that the graph of (F), (G(F) = {(x, F(x)) : x \in X}), is closed in (X \times Y). The goal is to show that (F) is continuous. i) Use of (X) being a Banach space: We define a sequence ((x_n)) in (X) such that (x_n \to x) in (X) and (F(x_n) \to y) in (Y). To conclude that (x) actually belongs to (X) (so the limit exists in (X)) and to apply limits inside the space, we rely on the completeness of (X). If (X) were not complete, the limit (x) of a Cauchy sequence might not exist in (X), and the argument would fail. Thus, the Banach property of (X) ensures the sequences we work with converge inside (X). ii) Use of (Y) being a Banach space:

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