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.

The closed graph theorem is a fundamental result in functional analysis, particularly in the context of Banach spaces. Let's analyze how each of the mentioned facts is used in the proof:

i) X is a Banach space:
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