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I'm attempting to prove the following:

If $X$ is a Banach space and $Y$ a normed vector space, then a bounded linear operator $T : X → Y$ is invertible if and only if $T$ and $T^*$ are bounded below.

I was under the impression that $T^*$ was the Hilbert-adjoint operator, but that doesn't seem to fit in this context. Any advice for how to proceed?

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