1
$\begingroup$

Let $X, Y$ be Banach spaces. I'm considering a bounded linear functional $g:X\to Y$ and its lift $g_\sharp: \mathcal{P}(X)\to \mathcal{P}(Y)$.

I want to consider the inverse of $g_\sharp$ in some sense, but I'm not sure what space and metric to consider for $g_\sharp$. My intuition is to use $\mathcal{L}(\mathcal{M}(X),\mathcal{M}(Y))$, where $\mathcal{M}$ is used to denote $\sigma$-finite measures. It also doesn't seem clear that the inverse will be $\mathcal{P}(Y)\to\mathcal{P}(X)$.

Does anyone have any familiarity with this or have any references?

Improve this question
$\endgroup$
2
  • $\begingroup$ What do you mean by "inverse in some sense"? The map $g_\sharp$ is in general neither injective nor surjective. $\endgroup$
    – Kostya_I
    Commented May 8 at 8:00
  • 1
    $\begingroup$ in my application, essentially I want some necessary conditions on $g$ so that: if $g_\sharp$ is invertible (in some embedded space), then $\operatorname{ran} g_{\sharp| \mathcal{P}(Y)}^{-1} = \mathcal{P}(X)$. Point is I'm not sure what to use for the embedding space and whether this restriction is meaningful. @Kostya_I $\endgroup$
    – optimal_transport_fan
    Commented May 8 at 9:50

0

Reset to default