2

I'm looking for an example of a linear functional $u: C_c^\infty(\Omega) \to \mathbb C$ ($\Omega \subset \mathbb R^n$ open), which is not a distribution. I could not find anything... I thought of something like taking the function $\frac 1 x$ and defining $u$ as $$\langle u, \phi \rangle = \int_{\Omega} \frac 1 x \phi(x) \, \mathrm dx \quad \text{for } \phi \in C_c^\infty(\mathbb R) \; ,$$ but this does not seem to work, because the integral is not defined for all $\phi \in C_c^\infty(\mathbb R)$. This is also the reason why we introduce the principle value of $\frac 1 x$. Does anybody know an example?

aexl
  • 2,102
  • 1
    Explicit examples are hard (if not impossible). The topology on the space of test functions is so fine that pretty much everything one can explicitly write down is continuous. – Daniel Fischer Feb 23 '16 at 13:37
  • I doubt $\displaystyle\sum_n \frac{R^n}{n!} \frac{d^n \delta(x)}{dx_i^n} $ is continuous, because it is well-defined only for analytic test functions whose radius of convergence around $0$ is $> R$. and $\displaystyle\sum_n \frac{d^n \delta(x)}{dx_i^n} $ or $\displaystyle\sum_n e^{e^{e^{n!}}} \frac{d^n \delta(x)}{dx_i^n} $ should be defined only for test functions being locally polynomials at $0$. – reuns Feb 23 '16 at 14:59
  • I don't understand what you mean with "is not defined for all $\phi$". if it is continuous somewhere, then it is continuous everywhere or it is unbounded (and not defined somewhere). the only alternative is continuous nowhere, as the non continuous solution of $f : \mathbb{R}\to \mathbb{R}$, $f(x+y) = f(x)+f(y)$ being continuous only on $\mathbb{Q}$ (but requiring the axiom of choice) – reuns Feb 23 '16 at 15:04
  • @user1952009 I think the question is about a linear functional defined everywhere, yet not continuous. An example of a linear functional which not defined everywhere is, for example, $T=\sum_{k\ge 0}\delta_0^{(k)}$. – TZakrevskiy Feb 23 '16 at 19:05
  • @TZakrevskiy : $T$ is defined on a dense subset so in some sense it is defined everywhere... I don't see your point, and he talked about $\int_A \frac1x \phi(x) dx$ which is the same : defined only for a dense subset of test functions (and can be extended to a distribution). what I meant is that among those $T_{a} = \sum_k a_k \delta^{(k)}$ there should be many which cannot be extended to a distribution – reuns Feb 23 '16 at 19:56

1 Answers1

2

You can concoct an example using a Hamel basis like in Discontinuous linear functional. Maybe this are the only possible examples like in the case of Banach spaces.

Community
  • 1
Martín-Blas Pérez Pinilla
  • 41,546
  • 4
  • 46
  • 89