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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?

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

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