Topologies on space of compactly supported continuous functions

Question

Let $X$ be a locally compact Hausdorff space. As far as I understand, the space $C_c(X) = C_c(X; \mathbb{C})$ of compactly supported continuous complex-valued functions on $X$ is (most?) often topologized as the limit of the spaces $C_K(X)$, where $$C_K(X) := \{ f\,\colon X \to \mathbb{C} \text{ continuous}, \, \mathrm{supp}(f) \subseteq K \}$$ and $K$ ranges over the compact subsets of $X$. However, it would also seem somewhat natural to consider $C_c(X)$ as a subspace of the space $C(X) = C(X; \mathbb{C})$ of all continuous complex-valued functions on $X$, which carries its own natural topology, namely the compact-open topology. My question is: are these two topologies secretly the same? (A reference would be great, but a sketch of an argument would of course also do.)

Also on this note, let $\mu$ be a nonzero Radon measure on $X$ (by which I mean a countably additive positive measure defined on the Borel-$\sigma$-algebra of $X$, finite on compact sets, inner regular on open sets and outer regular on Borel sets; the case I'm ultimately interested in is: $X = G$ is a locally compact group and $\mu$ is Haar measure). Then I'm given to understand that $C_c(X)$ is a dense subspace of $L^2(X) = L^2(X, \mu; \mathbb{C})$. Does the subspace topology coming from the inclusion $C_c(X) \hookrightarrow L^2(X)$ agree with either of the two topologies mentioned above? I cannot seem to find an answer, either by myself or with the help of a search engine, but this sounds like something which should be well-known (or maybe it's trivially true/false and I'm just not seeing it).

Answer 1

The example $X = \mathbb N$ (with Radon measure $\mu =$ counting measure) suggested by Abdelmalek Abdesselam indeed shows that the three topologies on $C_c(X)$ are not the same in general. Here are the details I have worked out (hopefully correctly):$\newcommand{\abs}[1]{|#1|}$ $\newcommand{\norm}[1]{\|#1\|}$

• the topology $\tau_2$ induced by the inclusion $C_c(\mathbb N) \hookrightarrow L^2(\mathbb N) = \ell^2(\mathbb N)$ is finer than the topology $\tau_{\rm co}$ coming from the inclusion $C_c(\mathbb N) \hookrightarrow C(\mathbb N)$. Indeed, a basic neighbourhood of $0$ in the latter topology is of the form $$V = V(n_1, \dotsc, n_r ; \varepsilon_1, \dotsc, \varepsilon_r) = \{f : \abs{f(n_i)} < \varepsilon_i, i = 1, \dotsc, r\}$$ where the $n_i$ are some natural numbers and the $\varepsilon_i$ are positive real numbers. If $\varepsilon > 0$ is smaller than the finitely many $\varepsilon_i$'s, then $V$ contains $\{f \in C_c(\mathbb N) : \norm{f}_2 < \varepsilon\}$ because each $f$ in the latter set satisfies $\abs{f(n)} < \varepsilon$ for all $n \in \mathbb N$.
  To see that the two topologies are not the same, note that on $V$ (for any choice of $n_i$ and $\varepsilon_i$), the $L^2$-norm is unbounded (because there is no control over all the infinitely many natural numbers outside of $\{n_1, \dotsc, n_r\}$), i.e., no $\tau_2$-open ball around $0$ fully contains a $\tau_{\rm co}$-neighbourhood of $0$.

• the inductive limit topology $\tau_{\rm lim}$ on $C_c(\mathbb N)$ is finer than $\tau_2$. Indeed, a fundamental neighbourhood of $0$ in the former topology is of the form $$W = W((\varepsilon_n)_{n \in \mathbb N}) = \{f : \abs{f(n)} < \varepsilon_n ~ \forall n\}$$ for some sequence $(\varepsilon_n)$ of positive real numbers. So for instance, for any $\varepsilon > 0$, the fundamental $\tau_2$-neighbourhood $\{f \in C_c(\mathbb N) : \norm{f}_2 < \varepsilon\}$ of $0$ contains $W((\varepsilon_n))$ with $\varepsilon_n = \varepsilon a_n$ where $(a_n)$ is any square-summable sequence of (strictly) positive real numbers with $\sum a_n^2 \le 1$.
  To see that the two topologies are not the same, observe that, whenever $(\varepsilon_n)$ is a sequence s.t. $\varepsilon_n \to 0$ for $n \to \infty$, the $\tau_{\rm lim}$-neighbourhood $W((\varepsilon_n))$ cannot contain any $\tau_2$-open ball $B_\varepsilon(0)$: indeed, such a ball always contains all the functions $$n \mapsto \begin{cases} \varepsilon, & \text{if } n = m, \\ 0, & \text{else} \end{cases} \qquad (m \in \mathbb N),$$ but $W((\varepsilon_n))$ will only contain finitely many of these functions because $\varepsilon_n$ becomes smaller than $\varepsilon$ eventually.

Answer 2

A bit long for a comment so I will post it as an answer.

I did not think of $C_c(\mathbb{N})$ as $\mathscr{D}(M)$ for some zero-dimensional manifold $M$ (with countably many connected components), but now that you mention it, yes that is exactly it. I like to think of $C_c(\mathbb{N})$ as a "bare bones" toy model for $\mathscr{D}(\Omega)=C_{c}^{\infty}(\Omega)$. A key difficulty that beginniners in the subject have to overcome is to understand why convergent sequences of test functions must be supported in a common compact set. Once this hurdle is cleared, you will become more comfortable with these funny spaces and they will become your friends. The easiest way to understand the issue is to treat the elementary case $C_c(\mathbb{N})$ first.

A convergent sequence form a bounded set and this last property is why you must have a common compact support. Recall that $A\subset C_c(\mathbb{N})$ is bounded iff for all continuous seminorm $\rho:C_c(\mathbb{N})\rightarrow [0,\infty)$, $$ \sup_{f\in A}\rho(f)<\infty\ . $$ Arguing by contradiction, suppose there is no common compact (i.e., here finite) support for the elements of $A$. Then one can construct a sequences $f_k\in A$ and $n_1<n_2<\cdots$ in $\mathbb{N}$, such that $f_k(n_k)\neq 0$ for all $k$. Recall that the natural topology of $C_c(\mathbb{N})$ is defined by the seminorms $$ \|f\|_{\varepsilon}=\sum_{n\in \mathbb{N}}\varepsilon_n|f(n)| $$ indexed by all sequences $\varepsilon\in [0,\infty)^{\mathbb{N}}$. Now let us construct such a sequence by letting $\varepsilon_n=k\times |f_k(n_k)|^{-1}$ if $n=n_k$ and otherwise let $\varepsilon_n=0$. Clearly $\|f_k\|_{\varepsilon}\ge k$, and so $\sup_{f\in A}\|f\|_{\varepsilon}=\infty$.

Now for $\mathscr{D}(\Omega)$, take a sequence of points $x_n$ that escape to the boundary of $\Omega$. Build some small disjoint balls around these points, and construct some bum functions $\phi_n$ supported inside these balls. The map $f\mapsto \sum_{n}f(n)\phi_n$ gives you a way to embed $C_c(\mathbb{N})$ inside $\mathscr{D}(\Omega)$ and also transfer counterexamples from one setting to the other.