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.
In the proof of the Closed Graph Theorem, several key facts are used to establish the continuity of a linear operator FF. Let's analyze where and how each of the mentioned facts is incorporated:
i) X is a Banach space:
This fact is crucial in ensuring that the space XX is complete, which plays a pivotal role in the proof. The Closed Graph Theorem requires that the domain space XX is complete because this ensures that sequences of Cauchy points in XX converge to an _________ _______ _______ __________ ________ ________ ___ ________ _______ ________ _______ ________.
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Let X be an inner product space and x, y ∈ X. Prove that x | y if and only if
Let {An} be a sequence of unitary operators in BL(H). Prove that if , then A is unitary.
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Give one example of each of the following. Also justify your choice of example.
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ii) A normal operator on a Hilbert space which is not unitary.
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is not uniformly bounded, then there exists a dense subset D of X such that for every
is not bounded in Y.
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