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My professor wants us to prove that every linear mapping from a finite dimensional Banach space is continuous.

BUT, he wants us to do so using the Closed Graph Theorem (in functional analysis). However, I just don't see how to do this... It actually seems way harder to use the Closed Graph Theorem instead of doing something similar. Does anyone have any hints?

Thanks in advance.

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If $T:X\to Y$ is a linear operator with graph $G$, then note that $G$ is a vector subspace of $X\oplus Y$. Using the fact that $X$ is finite dimensional try to convince yourself that $G$ is finite dimensional and hence closed. Now, use the Closed Graph Theorem.

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