Let $X$ and $Y$ be topological linear spaces which are complete & Hausdorff, and admit dual spaces which separate points. Suppose the topologies are non-separable and non-metrizable.

Let $f : X \to Y$ be a continuous linear map, and let $\Gamma_f$ be the space of (partial) sections of $f$. i.e., a continuous linear function $g : D \to Y$ defined on a (closed) domain $D \subseteq Y$ satisfying $f \circ g = I_D$.

The space $\Gamma_f$ always contains the zero section, defined on $D = \{0\}$. Is it possible for $\Gamma_f$ to contain just this one map, or must $\Gamma_f$ always contain non-trivial examples?

Is there a natural topology on $\Gamma_f$ which is finer than the weak topology?

  • 1
    $\begingroup$ Since $Y= Im(I_Y) = Im(f\circ g)\subseteq Im(f)$, $f$ is surjective, and there are counterexamples, like $f=I_X$. $\endgroup$ – Peter Michor Dec 19 '13 at 2:47
  • $\begingroup$ Thank you, @Peter. I've modified my question to avoid this trivial example. $\endgroup$ – Tom LaGatta Dec 19 '13 at 21:41

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.