\mathbb{R}^P$ are given by $$\begin{aligned}
    \partial_\mu a_\mu(u, v) \cdot \nu &= \sum_{\xi = 1}^{\Xi^a} \big(\partial_\mu \theta_\xi^a(\mu) \cdot \nu\big)\, a_\xi(u, v)&&\text{and}&&&
    \partial_\mu l_\mu(v) \cdot \nu &= \sum_{\xi = 1}^{\Xi^l} \big(\partial_\mu \theta_\xi^l(\mu) \cdot \nu\big)\, l_\xi(v),
    \notag
    \end{aligned}$$ respectively, if $u, v$ do not depend on $\mu$. We also introduce the following shorthand notation for the derivative of functionals and bilinear forms w.r.t. the parameter in the direction of $\nu \in \mathbb{R}^P$, e.g. for $\mu \in {\mathcal{P}}$ we introduce $$\begin{aligned}
    \partial_\mu l_\mu &\cdot \nu \in V'&&& v \mapsto \big(\partial_\mu l_\mu \cdot \nu\big)(v) &:= \partial_\mu l_\mu(v)\cdot \nu&&\text{and}
    \notag\\
    \partial_\mu a_\mu &\cdot \nu \in V \times V \to {\mathbb{R}}&&& u, v \mapsto \big(\partial_\mu a_\mu \cdot \nu\big)(u, v) &:= \partial_\mu a_\mu(u, v)\cdot \nu,
    \notag
    \end{aligned}$$ and note that $\partial_\mu l_\mu$ and $\partial_\mu a_\mu$ are continuous and separable w.r.t. the parameter, owing to Assumption [\[asmpt:parameter\_separable\]]{}.

The bilinear form $a_\mu(\cdot\,,\cdot)$ is continuous and coercive for all $\mu\in{\mathcal{P}}$. Thus we can define the bounded solution map $\mathcal{S}:{\mathcal{P}}\to V$, $\mu \mapsto u_\mu:= \mathcal S(\mu)$, where $u_\mu$ is the unique solution to for a given $\mu$. The Fréchet derivatives of $\mathcal{S}$ are a common tool for RB methods and optimization, e.g., for constructing Taylor RB spaces that consist of the primal 