This might be more of a physics question, but it is mathematics-related, I hope I am not out of place with this.
Let $(M,\mathcal{S},g)$ be a smooth, $n$-dimensional manifold equipped with a Riemann metric. Let us denote the vector space of $(p,q)$-type tensor fields on $M$ as $\mathcal{T}_{q}^{p}(M)$.
Let $\Psi:\mathbb{R}\rightarrow\mathcal{T}_{q}^{p}(M),\varepsilon\mapsto\Psi(\varepsilon)$ be a smooth curve and let us use the notation where $\Psi$ denotes $\Psi(0)$.
Let $S:\mathcal{T}_{q}^{p}(M)\rightarrow\mathbb{R}$ be a functional, in such way, that $$S[\Psi]=\int_{M}\mathcal{L}(\Psi,\nabla\Psi)\sqrt{|\det(g)|}\mathrm{d}x^{1}\wedge...\wedge\mathrm{d}x^{n}.$$
In this case, we say $S$ is functionally derivable at $\Psi$, if there exists a $\frac{dS[\Psi]}{d\Psi}\in\mathcal{T}_{p}^{q}(M)$ tensor field, that $$\left.\frac{dS[\Psi(\varepsilon)]}{d\varepsilon}\right|_{\varepsilon=0}=\int_M\frac{dS[\Psi]}{d\Psi}\bullet\left.\frac{d\Psi(\varepsilon)}{d\varepsilon}\right|_{\varepsilon=0}\sqrt{|\det(g)|}\mathrm{d}x^{1}\wedge...\wedge\mathrm{d}x^{n},$$ where $\bullet$ denotes full contraction.
My questions are regarding technical details of this derivative. Physics books generally do not impose rigorous conditions on the space of tensor fields on which $S$ is defined.
What structures does this space need to possess for this to make sense? I assume Hausdorff-topology is a must, but does it need to be normed? If so, what norm do we use, that does not conflict with physics?
Wald mentions in a footnote, that in general, a tensor distribution needs exist, so that $$\left.\frac{dS[\Psi(\varepsilon)]}{d\varepsilon}\right|_{\varepsilon=0}=\frac{dS[\Psi]}{d\Psi}\left[\left.\frac{d\Psi(\varepsilon)}{d\varepsilon}\right|_{\varepsilon=0}\right].$$ Is there any conceivable situation within the bounds of physics, where this distribution is NOT regular?