In Murphy's book it is stated that
''Observe that every invertible linear map is bounded below, as is every isometric linear map.''
It's clear to me that isometries are bounded below but I'm doubtful about invertible linear maps. I can't seem to prove it.
Should the claim really be ''every continuous invertible linear map between Banach spaces is bounded below''? Because this claim I can prove.
Edit
What would be an example of an invertible linear map between Banach spaces that is not bounded below?
Best Answer
The meaning of invertible depends on context. In the context of bounded linear operators, one says that $T$ is invertible if both $T$ and $T^{-1}$ are bounded linear operators. (In particular, $T$ is injective and onto, so that $T^{-1}$ is defined.)
Since $T^{-1}$ is bounded, it follows that $T$ is bounded below.