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

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.

A quotient operator is a surjective bounded linear operator that is an open map. For a linear map, being an open map is equivalent to the existence of $c>0$ such thatThe inclusion gives a neighbourhood of $Tx$ inside the image of every sufficiently small ball about $x$, by translation and scaling. Conversely, an open map sends $B_X(0,1)$ to a neighbourhood of zero. A linear open map onto $Y$ is automatically surjective, since its image contains a ball and is a linear subspace.

The Banach-space adjoint is $T^{\ast }g=g\circ T$. For $g\in Y^{\ast }$, the ball inclusion givesAlso $\Vert T^{\ast }g\Vert \le \Vert T\Vert \Vert g\Vert$. Thus the Banach-space adjoint is an isomorphic embedding, with lower bound $c$.