I'm having trouble in distribution theory, though not in the usual setting. The context is in theoretical physics, trying to solve BF theory.
My goal is to solve the following equation: $$\forall i \in \{1,...,n\}, (\chi(g_i)-2)f(g_1,...g_n) = 0$$ Where $f$ is a square integrable (with respect to the Haar measure) function over $SU(2)^n$ (actually an equivalence class of almost everywhere equal functions) and $\chi$ is the character in the fundamental representation (so basically the trace in $SU(2)$).
It doesn't take much to see that this equation has only one solution $f=0$ as $\chi$ is different from $2$ almost everywhere. So, the natural question is to extend the equation and try to solve it in distribution space.
It is fairly easy to see that the Dirac delta is a solution, the harder point being to prove that it is the only one.
My problem is at least two-fold:
- Is the solution indeed unique? I'm quite aware that the relevant property here is the continuity of distributions on the test space. But I didn't manage to use it effectively. But this is mostly related to my second problem, which is:
- What topology should we put on the space of test functions? The natural space of test functions would be, I think, the smooth functions over $SU(2)^n$. But, from my research, I've come to understand that the topology is induced as a descending intersection of the space $\mathcal{C}^k$ of k-continuous functions endowed with the natural norm to make the derivative continuous (as seen in these notes for instance or seemed to be implied in this question). Maybe here, it's my training as a physicist (rather than a mathematician) that bothers me, but I've trouble finding even a course explaining how such a topology is constructed, if it is metrizable, and if there is an explicit writing of the norm. I have the vague intuition that $\varphi \rightarrow 0$ in the topology is equivalent to $\forall \alpha, \|\partial_\alpha \varphi\|_\infty \rightarrow 0$ where $\alpha$ is a multi-index. Is this even remotly true?
Assuming this last, perhaps false, intuition on the topology, I have tried two courses to prove my result.
The first idea was to use sequential continuity (which should at least be implied by continuity) and try to find for every test function $\varphi$ that cancels on $\mathbb{1}$ (the identity) a sequence $\varphi_n$ such that $T(\varphi_n) = T(\varphi)$ and $\varphi_n \rightarrow 0$ where $T$ is a distribution solution of the equation. Though, it is quite easy to find similar sequences that converge absolutly, I didn't manage to control the convergence of the derivatives.
A second idea was to use the continuity directly and try to show that the image of the multiplication by $\chi_i-2$ is dense in the kernel of $\delta$. This is actually false when $n\neq 1$ but we can adapt the proof, I think, and just try to show that the image of the multiplication by $\chi_i - 2$ is dense in the space of functions that cancel when $g_i = \mathbb{1}$. But once again, I couldn't find sequences that converge in the good sense.
Sorry for my mathematical naivety. But I would be grateful for good references on the topology of the space of test functions. Or, maybe, did I just miss some obvious way to solve the problem?