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 continous function of $f(t)$. Then let $g:L^2(\mathbb{R}^d,\mathbb{R}) \rightarrow L^2(\mathbb{R}^D,\mathbb{R})$ be a continous map.
Does it follow that the functional
$$
F^{\star}(g) \triangleq F(g(f)),
$$
is continuous in $g$?
Composition of integral operator continous
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0$F : L^2 \to \mathbb R$ and $d=D$? – 2017-01-19
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0No in general I do not want $d=D$ unless needed. – 2017-01-19
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0Is $F : L^2(\mathbb R^D , \mathbb R) \to \mathbb R$? – 2017-01-19
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0Yes. Infact it is into $[0,\infty)$ – 2017-01-19
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0Indeed your $F^*$ still does not make sense, you mixed up $d$ and $D$. – 2017-01-20
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0Thanks for pointing that out, I make left right errors from time to time – 2017-01-20