It's clear, that in the finite dimension there is the identity matrix which has this feature:
$$ AA^{-1} = A^{-1}A = I $$
I know, that it's not supposed to be generally true in the normed linear spaces of the infinite dimension.
So, could you, please, provide me any example of normed linear space and continuous linear mappings $f$ and $g$ which have the following feature?
$$ f \circ g = I\\ g \circ f \neq I $$
I'm not very experienced with this, so I really can't think of any example right now.
The canonical example is given by the unilateral shift. You take $X=\ell^2(\mathbb N)$, and consider $$ S(a_1,a_2,\ldots)=(0,a_1,a_2,\ldots) $$ and $$ T(a_1,a_2,a_3,\ldots)=(a_2,a_3,\ldots). $$ Then $TS=I$ but $$ ST(a_1,a_2,\ldots)=(0,a_2,a_3,\ldots). $$ Both $S,T$ are bounded with $\|S\|=\|T\|=1$. In fact, $S$ is an isometry: $\|Sa\|=\|a\|$ for all $a$. It is an example of what is called a proper isometry.