I'm writing some notes on differential calculus over normed spaces and I'm having some trouble finding the appropriate notation for partial derivatives. Here's what I mean:
Derivative of a map between normed spaces. If $f:E\to F$ is a map between normed spaces $E$ and $F$ over the same field $\mathbb{K}$, I say it is differentiable at $a\in E$ if there exists a continuous linear map $T\in\mathcal{L}(E,F)$ such that $$\lim_{h\to 0}\frac{\lVert{f(a+h)-f(a)-Th}\rVert_F}{\lVert{h}\rVert_E}=0.$$ It is easy to see that if $T$ exists then it is unique. I call it the derivative of $f$ at $a$, and denote it by $\mathrm{d}_a f$. The reason I choose to use this notation and not the (more common) $\mathrm{D}f(a)$ is that, since the derivative is itself a map, it can take inputs $v\in E$, so in the second case I would have to write the more awkward $\mathrm{D}f(a)(v)$ instead of the more comfortable $\mathrm{d}_a f(v)$ for the image of $v$ under $\mathrm{d}_a f$. See this answer from Qiaochu Yuan for more context.
Partial derivatives. Let $E=E_1\times\cdots\times E_n$ be a product of normed spaces and let $F$ be another normed space. Let $f:E_1\times\cdots\times E_n\to F$ be a map, which can be written as $(x_1,\ldots,x_n)\mapsto f(x_1,\ldots,x_n)$, where $x_i\in E_i$ for $i=1,\ldots,n$. Let $a=(a_1,\ldots,a_n)\in E$ be fixed. If the map \begin{align*} E_r&\to F\\ x_r&\mapsto f(a_1,\ldots,a_{r-1},x_r,a_{r+1},\ldots,a_n) \end{align*} is differentiable, I say that its partial derivative with respect to to the $r$th variable at the point $a$ is the derivative of the above map at the point $a_r$.
Question. As I said, I can't decide what notation to use here for partial derivatives. Some authors use $\mathrm{D}_r f(a)$ or $\partial_r f(a)$ to denote it, but we run into the same "problem" I mentioned before (the partial derivatives here are itself linear maps). The most comfortable notation I have found so far is $\partial_r f\rvert_a$, but it still does not convince me, because I want to have a very similar notation for the case $E_1=\cdots=E_n=\mathbb K$, where we get the classic partial derivative by identifying the derivative with its value at $1\in\mathbb K$: $\partial_r f(a) = (\partial f/\partial x_r)(a) = \partial_r f\rvert_a (1)\in F$.
Any help or suggestion is very appreciated.
There are a few standard notations for partial derivatives of maps between normed spaces:
Df(a)(v) or Df(a).v for the total derivative at a, as a linear map. ∂rf(a) or ∂f/∂xr(a) for the partial derivative with respect to the rth variable. ∂rf|a to emphasize that this is the derivative of the "restricted" map xr ↦ f(a1, ..., ar−1, xr, ar+1, ..., an).
So for your example, I would write: ∂f/∂xr(a) = ∂rf|a(1)
The notation ∂rf|a has the advantage that it's consistent whether xr ranges over K or a general normed space Er. The notation ∂f/∂xr(a) is more standard when xr ranges over a scalar field (like R or C), but is a bit inconsistent when Er is a normed space.
Either notation is fine, so I would choose based on what you're emphasizing or what your audience would find most familiar.