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

We use successive approximation for quotient operators. Fix $y$ with $\Vert y\Vert <1$. The hypothesis gives $x_0$ with $\Vert x_0\Vert <1$ and $\Vert y-T{x}_0\Vert <\delta$. Inductively, ifapply the hypothesis to $r_n/{\delta }^n$, obtaining $x_n$ with $\Vert x_n\Vert <{\delta }^n$ and $\Vert r_n-T{x}_n\Vert <{\delta }^{n+1}$. If a residual is zero, take all remaining terms zero.

The series $\sum x_n$ converges in the Banach space $X$ because $\sum \Vert x_n\Vert <\sum {\delta }^n=(1-\delta {)}^{-1}$. Write its sum as $x$. The continuity of $T$ and the residual estimate giveScaling now yieldsThis ball inclusion proves that $T$ is a quotient operator. Completeness of $Y$ was unnecessary: the convergent series is constructed in $X$ and its image has limit $y$.