Past exam of the functional Analysis and Banach and C*-Algebras course of the University of Oxford / 2010 / A / 1 / i / a

An isomorphic embedding of normed vector spaces is a bounded linear operator that is injective and whose inverse function on its image is bounded. Equivalently, $T$ is bounded below: there is $c>0$ such thatIndeed this inequality gives injectivity and $\Vert T^{-1}z\Vert \le c^{-1}\Vert z\Vert$ for $z\in T(X)$; the converse follows by applying the boundedness of $T^{-1}$ to $Tx$. Together with $\Vert Tx\Vert \le \Vert T\Vert \Vert x\Vert$, it makes $T$ an isomorphic embedding onto its image with the inherited norm. The image is necessarily closed when $X$ is a Banach space; that extra conclusion requires completeness.