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Everything below is from Rudin's Functional Analysis .

  • $\Omega$ is an open subset of $\mathbb{R}^n$.
  • $\mathcal{D}(\Omega)$ is the space of test (smooth, compactly supported) functions $\Omega\to\mathbb{C}$ with a complete, unmetrizable topology $\tau$.
  • $\mathcal{D}_K(\Omega)$ is the space of test functions supported on the compact set $K\subseteq \Omega$, with a Fréchet-space topology $\tau_K$ that corresponds with the subspace topology under $\tau$.
  • $\mathcal{D}'(\Omega)$ is the space of distributions i.e. of continuous linear functionals $\mathcal{D}(\Omega)\to\mathbb{C}$.

Theorem 6.8: if $\Lambda$ is a linear functional on $\mathcal{D}(\Omega)$, then the following conditions are equivalent:

  1. $\Lambda\in\mathcal{D}'(\Omega)$, meaning $\Lambda$ is a distribution.
  2. To every compact $K\subseteq\Omega$ corresponds a nonnegative integer $N$ and a constant $C<\infty$ such that $$|\Lambda\phi|\le C\|{\phi}_N\|$$ holds for every $\phi\in\mathcal{D}_K$.

Proof: This is precisely the equivalence of (a) and (d) in Theorem 6.6, combined with the description of the topology of $\mathcal{D}_K$ by means of the seminorms $\|\phi\|_N$ given in Section 6.2.

I struggle to understand the proof. Here is what I do understand:

It is sufficient to show that condition $2)$ is equivalent to d) in Theorem 6.6, which reads:

$$\text{d) The restriction of $\Lambda$ to any $\mathcal{D}_K(\Omega)$ is continuous.}$$

Now, as $\mathcal{D}_K(\Omega)$ is metrizable, such a thing happens if and only if $\Lambda|_{\mathcal{D}_K(\Omega)}$ is bounded, which seems similar (though not that I can tell equivalent) to condition $2)$.

In case it is of any help, the seminorms $\|\phi\|_N$, which provide $\mathcal{D}_K(\Omega)$ with its topology, are defined as $$\|\phi\|_N:\mathcal{D}_K(\Omega)\to\mathbb{R}:\phi\mapsto \sup\{|D^\alpha \phi(\textbf{x})|:\textbf{x}\in\Omega, |\alpha|\le N\}.$$

Sam
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1 Answers1

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There is a useful characterisation of continuous linear functionals on a locally convex space which will be of use here. The statement is as follows.

Let $X$ be a locally convex space whose topology is generated by a family of seminorms $\{p_{\alpha} : \alpha \in \mathcal{A}\}$. Then a linear functional $f$ on $X$ is continuous if and only if there is a finite subset $F$ of $\mathcal{A}$ and positive scalars $\{c_{\alpha} : \alpha \in F\}$ such that

$$|f(x)| \leq \sum_{\alpha \in F} c_{\alpha} p_{\alpha}(x)$$

for all $x\in X$.

To briefly explain why this characterisation is valid, note that a linear functional $f$ on a locally convex space $X$ is continuous if and only if the seminorm $p$ on $X$ defined by $p(x) := |f(x)|$ is a continuous seminorm. The characterisation then follows from this result.

To apply the above result to your problem, note that by your remark $\Lambda$ is continuous on $\mathcal{D}(\Omega )$ if and only if given any compact subset $K$ of $\Omega$, it follows that the restriction of $\Lambda$ to $\mathcal{D}_{K}$ is continuous. Fix a compact subset $K$ of $\Omega$. Then by the above characterisation of continuous linear functionals on a locally convex space, this holds if and only if there is a finite subset $F$ of $\mathbb{N}$ and a collection of positive scalars $\{c_{k}: k\in F\}$ such that

$$|\Lambda (\phi )| \leq \sum_{k\in F}c_{k} \|\phi\|_{k}$$

for all $\phi \in \mathcal{D}_{K}$. By taking $N := \max \{k : k \in F\}$ and $C := \sum_{k\in F}c_{k}$ in the case above, it follows that this is equivalent to the existence of some $N\in\mathbb{N}$ and $C\in (0, \infty )$ such that

$$|\Lambda (\phi )| \leq C\|\phi\|_{N}$$

for all $\phi \in \mathcal{D}_{K}$.

Dean Miller
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