Let $F:L^2(\mathbb{R}^D,\mathbb{R})\to \mathbb R$ be a functional of the form $$ F(f) \triangleq \int_{\mathbb{R}^D} E(f(t))dm(t), $$ where $E$ is a continuous function of $f(t)$. Then let $g:L^2(\mathbb{R}^d,\mathbb{R}) \rightarrow L^2(\mathbb{R}^D,\mathbb{R})$ be a continuous map.
Does it follow that the functional $$ F^{\star}(g) \triangleq F(g(f)), $$ is continuous in $g$?