I wanted to be sure about the following:
Let's say we have vector spaces normed spaces $X$ and $Y$ and a linear operator $T:X \rightarrow Y$.
My idea was to reduce the properties that I need to show of the following two types of linear operators:
Then T is a isomorphism (continuous and bijective) if we have that $T$ is continuous + surjective and $T^{-1}$ is continuous. Correct?
Afais $||T^{-1}x||\le ||T^{-1}||||x||$ should imply that $T$ is injective.Hence, $T$ is a bijection. Hence, $T$ is a isomorophism.
$T$ is an isometric isomorphism if $||Tx||=||x||$ and $T$ is surjective. Correct? Cause I think this implies that $T$ is injective and that $T^{-1}$ is also an isometric isomorphism.
Best Answer
If you want to be really sure about this, you have to also be sure you are using the word "inverse" correctly. There are ordinary inverses, left inverses and right inverses: two of these three imply (by their very existence) that $T$ is injective, but the last one does not.
If $T^{-1}$ means the ordinary inverse (as it should), then note that saying "$T^{-1}$ is continuous" contains a sub-statement: "$T^{-1}$ exists". This sub-statement implies the injectivity of $T$, without any norms entering the picture.