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Let $\{z\mapsto e_i(z)\}_{i\in \mathbb{N}}$ be an orthonormal basis of $L^2(\mathbb{R}^d,\mathbb{R})$ and define the map $\mathscr{F}$ be the map from $L^2(\mathbb{R}^d,\mathbb{R})$ into $\ell^2$ by $$ \mathscr{F}\triangleq f(x)\mapsto \left( \int_{\mathbb{R}^d} f(z)e_i(z) dm(z) \right)_{i \in \mathbb{N}}, $$ where $m$ is the Lebesgue measure on $\mathbb{R}^d$.

Then what would the Frechet derivatives of $\mathscr{F}$ be and when do they exist?

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    It is a linear operator, so $\frac{\partial}{\partial t}\mathscr{F}[f+t g] = ?$ – 2017-01-09
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    So its it equal to itself? – 2017-01-09

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