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 with dense image, and let $$\Gamma_f := \big\{ g: Y \to X : f \circ g = I_Y \big\}$$ be the space of continuous linear sections of $f$.

Is $\Gamma_f$ necessarily empty? If not, what's a counterexample? Is there a natural topology on $\Gamma_f$ which is finer than the weak topology?

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?