Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 22F / a / Solution

The open mapping theorem states that every surjective bounded linear map $S:X\to Y$ between Banach spaces maps open sets to open sets. Equivalently, there is $c>0$ such that

The closed graph theorem states that a linear map $T:X\to Y$ between Banach spaces is continuous if and only if its graphis closed in $X\times Y$.

Assume the open mapping theorem and suppose $\Gamma (T)$ is closed. As a closed subspace of the Banach space $X\times Y$, the graph is itself Banach. The coordinate projectionis a bounded linear bijection. By the open mapping theorem, $P^{-1}$ is bounded. The other coordinate projection $Q:\Gamma (T)\to Y$ is bounded, andThus $T$ is bounded and hence continuous. This is the closed graph theorem from the bounded inverse theorem argument.

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