# Properties of the $L^2$ dual realizing test functions as distributions

Let $$U \subset \mathbb{R}^n$$ be open; let $$C^\infty_c(U)$$ denote the smooth real/complex compactly-supported functions on $$U$$, with the "Canonical LF" topology, i.e. the limit topology from all the restriction maps $$C_c^\infty(U) \rightarrow C^\infty(K)$$ for compact $$K\subset U$$. (I assume $$C^\infty(K)$$ has the the uniform topology.) Let $$\langle f,g\rangle = ||\bar{f}g||_{L^1(U)} = \int_{U}|dx|\,\overline{f(x)}\,g(x)$$ be the standard inner product. Let $$W = C^\infty_c(U)^*$$ be the strong dual of $$C^\infty_c(U)$$.

I have the following questions:

1. Is the linear injective map $$\phi : C^\infty_c(U) \rightarrow W : f \mapsto (g \mapsto \langle f,g\rangle)$$ also continuous? If so, is it an embedding of TVSs?

2. Is the image in $$W$$ of $$\phi$$ dense, for the weak-$$\ast$$ topology (for $$\langle\cdot,\cdot\rangle$$) on $$W$$? What about for the strong dual topology on $$W$$?

I am trying to examine these but unraveling the limit topology on $$C^\infty_c(U)$$ and the strong dual topology on $$W$$ is somewhat complicated. Does anyone know any techniques that can be applied here?

\begin{align*} \phi\text{ c'ts.}&\Leftrightarrow(\forall f_n\to0\in C_c^{\infty}(U))(\phi(f_n)\to0\text{ in the strong dual topology})\\ &\Leftrightarrow(\forall f_n\to0)(\forall\text{ bounded }B\subseteq C_c^{\infty}(U))\left(\lim_n{\sup_{g\in B}{\phi(f_n)(g)}}=0\right) \end{align*}

$$C_c^{\infty}(U)$$ has no norm, so "bounded" here needs a little definition. Wikipedia jumps straight to duality, but the simplest definition is von Neumann boundedness: for all neighborhoods $$V$$ of the origin, there exists $$r$$ such that $$rB\subseteq V$$. To match the Wikipedia definition, $$B$$ should be von-Neumann bounded in the weak topology on $$C_c^{\infty}(U)$$.

Since the weak topology is coarser than the standard topology, we may pass to the latter a fortiori. In fact, we can always expand $$B$$ to a set of the form $$B=\{f:(\forall x)(|f(x)|\leq R)\}\subseteq C_c^{\infty}(U)$$

But then $$\sup_{g\in B}{\phi(f_n)(g)}=\sup_{g\in B}{\langle f_n,g\rangle}\leq R\|f_n\|_{L^1}$$ Since convergence in $$C_c^{\infty}$$ implies convergence in $$L^1$$, the latter tends to $$0$$. Thus $$\phi$$ is indeed continuous.

As to whether $$\phi$$ is an embedding, it depends on what you mean by embedding.

• $$\phi$$ is injective: if $$\phi(f)=0$$, then for all $$g\in C_c^{\infty}(U)$$, we have $$0=\phi(f)(g)=\langle f,g\rangle$$; taking $$g=f$$ shows that $$f=0$$.
• But $$\phi$$ is not bicontinuous. For each $$n$$, let $$f_n:C_c^{\infty}(\mathbb{R})$$; $$f_n(x)=\frac{1}{1+(x-n)^2}$$. By an argument given below, any bounded $$B\subseteq C_c^{\infty}(\mathbb{R})$$ is contained in $$C_c^{\infty}(K)$$ for some compact $$K$$, so that $$\phi(f_n)\to0$$ on $$B$$. But $$f_n\not\to0$$ as elements of $$C_c^{\infty}(U)$$, since a sequence converging to $$0$$ in $$C_c^{\infty}(U)$$ must have uniformly compact support.

The image is dense in both topologies; the strong dual topology is finer than the weak-$$*$$, so it suffices to prove density there.

First, I claim distributions of compact support are dense. To be precise, fix an enumeration $$\{K_n\}_{n=0}^{\infty}$$ of increasing compact sets exhausting $$U$$ and find a "partition of unity": for each $$n$$, let $$\psi_n\in C_c^{\infty}(U)$$ be smooth, $$1$$-bounded, and such that $$\psi_n|_{K_n}=1$$ but $$\psi_n|_{U\setminus K_{n+1}}=0$$. If $$v\in W$$, then I claim that the functionals $$v_n:W$$; $$v_n(f)=v(f\cdot\psi_n)$$ converge to $$v$$.

To see this, pick bounded $$B\subseteq C_c^{\infty}(U)$$. Suppose for contradiction that there exists $$\{x_n\}_{n=0}^{\infty}\in U^{\omega}$$ such that, for all $$n$$, there exists $$f_n\in B$$ with $$f(x_n)\neq0$$, but $$x_n\notin K_n$$. Then $$w:W$$; $$w(g)=\sum_n{f_n(x)g(x)}$$ is well-defined and continuous (since elements of $$C_c^{\infty}(U)$$ have compact support) but is unbounded on $$B$$. Thus $$B\subseteq C_c^{\infty}(K_n)$$ for some $$n$$ (by abuse of notation), so that, for $$m\geq n+1$$, $$v_m$$ and $$v$$ coincide on $$B$$.

Second, recall that distributions of compact support are the derivatives of functions; that is, if $$v\in W$$ has compact support, then there exists $$g\in C(U)$$ and a multiindex $$\alpha$$ such that, for all $$f$$, $$v(f)=\langle g,f^{(\alpha)}\rangle$$ This is theorem 6.26 in Rudin, Functional Analysis (1991), and takes some machinery to prove.

Lastly, pick a sequence of smooth functions converging to $$g$$ and integrate by parts; this gives a sequence from $$\operatorname{im}{\phi}$$ converging to $$v$$.

• Thank you so much for the explanations! Commented Jun 29, 2022 at 0:55
• Could I ask a question: at the end of proving 1, how were you able to conclude $\phi$ is an embedding? Commented Jun 29, 2022 at 1:05
• @IndraneelTambe: I assumed "embedding" just means "injective continuous map". Did you mean something stronger? Commented Jun 29, 2022 at 3:40
• Oh I see, I meant an embedding in the sense of being a topological embedding, so a homeomorphism onto its image. I probably should have clarified this in my question. Commented Jun 29, 2022 at 5:12
• @IndraneelTambe Ah, sorry. That makes more sense based on what you wrote. Then no, it's not an embedding; I've edited the answer to reflect as much. Commented Jun 29, 2022 at 7:49