CORRECTION: As Simon Henry points out in the comments, there is a problem in the construction: the maps $\varphi_n$ are not necessarily linear. Under some additional constraints on the space (e.g. $X$ metrizable) the (set-valued) Tietze extension map has a linear selection (see "Tietze Extensions and Continuous Selections for Metric Projections" by Deutsch, Li and Sung-Ho Park). This can be used to turn $\phi_n$ into a linear map or directly define such a linear map $\psi_n$. Maybe, there is another way (without Tietze extension) to get the maps $\psi_n$ with the intention that the locally convex inductive limit topologies on $C_c(X)$ coincide.

Let $X$ be locally compact Hausdorff and for compact $K \subseteq X$ consider the Banach spaces $C(K)$ of continuous functions on $K$ and $C(X; K)$ of continuous functions on $X$ with support in $K$ both equipped with the supremum norm. ($C(X; K)$ can be considered as a closed subspace of $C(K)$.) The union of all the $C(X; K)$ is the space $C_c(X)$ of compactly supported continuous functions on $X$ which can then be equipped with the locally convex inductive limit topology.

Now let $X$ be $\sigma$-compact in which case we have a sequence $K_n$ of compact sets that covers $X$. Then, $C_c(X)$ is the strict inductive limit of the sequence $C(X; K_n)$ and thus an LB space. Usually, this is the way how the inductive limit topology on $C_c(X)$ is constructed in the literature.

I try for an alternative construction for $\sigma$-compact $X$ by employing the spaces $C(K_n)$ instead of $C(X; K_n)$. From Engelking, Ex. 3.8.C we can choose $K_n$ to be such that $K_n$ is contained in the interior of $K_{n+1}$. Then the sequence $C(K_n)$ can be turned into an inductive system as follows. Since $X$ is $\sigma$-compact it is normal and we have Tietze's extension theorem which gives us for each $n \in \mathbb{N}$ an isometric embedding $\varphi_n : C(K_n) \to C(X; K_{n+1})$. (Take $f \in C(K_n)$, set $f = 0$ on the closed set $(int(K_{n+1}))^c$ and extend the so gained continuous function $f : K_n \cup (int(K_{n+1}))^c \to \mathbb{R}$ to a continuous function $\varphi_n(f)$ on $X$ such that $\varphi_n(f) \in C(X; K_{n+1})$ and $\sup_{x \in K_n} |f(x)| = \sup_{x \in X} |(\varphi_n(f))(x)|$.) The restriction $C(X; K_{n+1}) \to C(K_{n+1})$ is an embedding and we get an embedding $\psi_n : C(K_n) \to C(K_{n+1})$. These maps are extended to an inductive system $\psi_{n, n+k} : C(K_n) \to C(K_{n+k})$, $\psi_{n,n+k} = \psi_{n+k-1} \circ \dots \circ \psi_n$ and its limit equals $C_c(X)$.

Question 1: I have not met such a construction in the literature, which gives rise to the question: Is there some mistake in the proof?

Question 2: Is there also a possibility to characterize $C_c(X)$ by all the $C(K)$ for $K \subseteq X$ compact for general locally compact Hausdorff spaces $X$ (not necessarily $\sigma$-compact)? In this case, $X$ needs not be normal and we do neither have countable exhaustions of $X$ by compact sets nor the Tietze extension theorem.

  • $\begingroup$ I see not much difference in your construction. Let me say it this way: Since $C(K_n)$ isometrically embeds in $C(X;K_{n+1})$ (as you noted) the "original construction" is the strict inductive limit of the $C(X;K_n)$ while your construction is the strict inductive limit of the $C(K;K_{n+1})$… $\endgroup$ – Dirk Feb 26 '16 at 9:58
  • $\begingroup$ @Dirk Do you mean $C(X; K_{n+1})$? Your comment seems to answer question 1. For question 2: the existence of such a sequence $C(X; K_1) \subseteq C(K_1) \subseteq C(X; K_2) \subseteq C(K_2) \subseteq \dots$ seems to heavily rely on $\sigma$-compactness of $X$. $\endgroup$ – yadaddy Feb 26 '16 at 10:12
  • $\begingroup$ Am I missing something or the $\Psi_n$ are not linear maps ? then in what category are you trying to take the limits ? $\endgroup$ – Simon Henry Feb 26 '16 at 10:14
  • $\begingroup$ Oops, yes $C(X;K_{n+1})$ is what I meant. For question 2 I have no idea. $\endgroup$ – Dirk Feb 26 '16 at 10:18
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    $\begingroup$ There is another problem: The restriction $C(X,K_{n+1})\to C(K_n)$ is not an embedding. Thus, you need to consider locally convex inductive limits with non-injective linking maps. $\endgroup$ – Jochen Wengenroth Feb 26 '16 at 11:30

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