$j_\xi \in V'$ for $1 \leq \xi \leq \Xi^j$ and $k_\xi: V \times V \to {\mathbb{R}}$ for $1 \leq \xi \leq \Xi^k$, and respective parameter functionals $\theta_\xi^a, \theta_\xi^l, \theta_\xi^j, \theta_\xi^k: {\mathcal{P}}\to \mathbb{R}$, such that $$\begin{aligned}
    a_\mu(u, v) &= \sum_{\xi = 1}^{\Xi^a} \theta_\xi^a(\mu)\, a_\xi(u, v)
    , &&&&& l_\mu(v) &= \sum_{\xi = 1}^{\Xi^l} \theta_\xi^l(\mu)\, l_\xi(v),
    \end{aligned}$$ and analogously for $j_\mu$ and $k_\mu$.

Due to Assumption \[asmpt:parameter\_separable\], all quantities which linearly depend on $a_\mu$, $l_\mu$, $j_\mu$ and $k_\mu$ (such as ${\mathcal{J}}$ or the primal residual) are also separable w.r.t. the parameter. Since we will use a Lagrangian ansatz for an explicit computation of derivatives, we require some notation that we use throughout this paper.

A note on differentiability {#sec:differentiability}
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If ${\mathcal{J}}: V \times {\mathcal{P}}\to {\mathbb{R}}$ is Fréchet differentiable w.r.t. each $\mu \in {\mathcal{P}}$, for each $u \in V$ and each $\mu \in {\mathcal{P}}$ there exists a bounded linear functional $\partial_\mu {\mathcal{J}}(u, \mu) \in \mathbb{R}^P$, such that the Fréchet derivative of ${\mathcal{J}}$ w.r.t. its second argument in the direction of $\nu \in \mathbb{R}^P$ is given by $\partial_\mu {\mathcal{J}}(u, \mu) \cdot \nu$ (noting that the dual space of $\mathbb{R}^P$ is itself). We refer to $\partial_\mu {\mathcal{J}}(u, \mu)$ as the derivative w.r.t. $\mu$. In addition, 