Let $(X,\mathscr{B},\mu)$ be a $\sigma$-finite measure space. Let $\gamma$ be a probability measure on $L_2(\mu)$ with $\mathrm{supp} \, \gamma = L_2(\mu)$ and existing first moment. Then

$$ f \mapsto \|f\| := \int |\langle f,g\rangle|\,\gamma(\mathrm{d}g) $$

is a norm on $L_2(\mu).$ Does this norm have a name? Is $L_2(\mu)$ with that norm complete? Is it complete with a modified version like

$$ f \mapsto \|f\|' := \sqrt{\int |\langle f,g\rangle|^2\,\gamma(\mathrm{d}g)}, $$

where we assume that $\gamma$ has a finite second moment?

Can anyone give me some references where such normed vector spaces were studied?

Why is that interesting? And what makes me think that this or related stuff was already studied?

By $\|f \|$ and $\|f\|'$ we consider (somehow) the usual $p$-norm of $f$ under the coordinate transformation $f \mapsto (\langle f,g\rangle)_{g \in L_2}.$ Actually, if $F \subset L_2(\mu)$ separates points it might be sufficient to consider a coordinate transformation $f \mapsto (\langle f,g\rangle)_{g \in F}.$ Correspondingly, consider then a probability measure $\gamma$ on $F$ with $\mathrm{supp}\, \gamma = F$ and existing second moment. After all it seems to me that considering $\|\cdot\|'$ on $L_2(\mu)$ instead of $\|\cdot\|_2$ does not change much. In fact, I think that $(L_2(\mu),\|\cdot\|')$ is a Banach space, however, it is not trivial.