2 Major correction

Thanks to your other question, I was on a LCTVS kicktoday. I did find one general criterion that implies that a locally convex space is paracompact. According to the Encyclopedia of Mathematics, if it is Montel (which means that it is barrelled and the Heine-Borel theorem holds true for it), then it is paracompact. Although this criterion is important, it is of no use to your specific question.

On the other hand,

I have an answer to thought I had a proof for half of your specific question: , which I think that any direct sum of Frechet spaces wrote up as the first version of any cardinality is normalthis answer, but I made a mistake and proved something different. One way (maybe the way) to understand normality My thinking is based on the fact that it the normality axiom for a topological space is equivalent to the Tietze extension theorem. If you know that a space is constructed from other spaces that are normal, you can try to use (Tietze extension to confirm the follows from normalityaxiom in the new space. If In the other direction, if $A$ and $B$ are the two closed sets, you obtain disjoint open neighborhoods from a continuous function that is 0 on $A$ and 1 on $B$. Once you have confirmed the axiomB$.) However, in my argument I conflated the powerful Tietze theorem gives you general Tietze extensionslocally convex direct sum of spaces with the topological direct sum. Let$X$be For a Frechet space. (Actually countable direct sum of copies of$X$can be any pointed metrizable space\mathbb{R}$, they are the same topology, and they agree with the box topology. The marked point defines But Waelbroeck, LNM 230 points out that they are different in the coproduct operation of your question.) uncountable case.

Let $\alpha$ be an ordinal, for instance the an ordinal of a well-ordering of cardinality $\mathbb{R}$. 2^{\aleph_0}$. Then$X^\alpha$is normal by transfinite induction\mathbb{R}^\alpha$ in the topological direct sum topology satisfies Tietze extension. Let $A$ and $B$ A \subset \mathbb{R}^\alpha$be disjoint a closed sets in set and let$X^\alpha$; we hope f:A \to extend the \mathbb{R}$ be a continuous functionthat is $0$ on $A$ and $1$ on $B$. . For $\beta < \alpha$, let $A_\beta$ and $B_\beta$ be the intersections intersection of $A$ and $B$ with $X^\beta$. \mathbb{R}^\beta$. Suppose that$\alpha = \beta+1$is a successor ordinal. Since If$X^\beta$\alpha$ is normal finite, then the conclusion is standard. Otherwise, by induction, there is a Tietze an extension of the function that is $0$ on $A_\beta$ and $1$ on f_\beta$of$B_\beta$. It can be extended f$ to $\beta+1$ because $X^\beta \cup A_{\beta+1} \cup B_{\beta+1}$\mathbb{R}^\beta$. Moreover, because that is a closed set by induction in$X^{\beta+1}$. This establishes the extension for successor ordinals. Okay, this looks like a circular argument, but it's not. If$\beta$is a finite ordinaldifferent sense, then we have already proved that$X^{\beta+1}$is metrizable, so it \mathbb{R}^{\beta+1}$ is normal. If $\beta$ is an infinite ordinal, then actually since $\beta$ and $\beta+1$ have the same cardinality, so we have already shown that $X^{\beta+1}$ is normal and we can use the Tietze theorem on it. So there exists an extension $f_\alpha$ to $\mathbb{R}^\alpha$. If instead $\alpha$ is a limit ordinal, then the extensions all the way up to $\alpha$ work just because they work. In order to understand the limit ordinal case, it is helpful to realize that ; that's the behavior of topological direct limit limits.

Having failed to normality for direct sums in the category of locally convex spaces is just the box topology; in this special case it is the same as the direct limit in topological spaces. (Boxes happen to be convex.)

As for sum, I can't say much about paracompactness either. :-) However, there is an interesting result called the Michael selection theorem which seems to do for paracompactness what the Tietze theorem does for normality. The question is whether there If the Tietze theorem is a "Michael" version of useful for your spaces, then maybe the above argument. I would not be surprised if there Michael selection theorem is too.

1

Thanks to your other question, I was on a LCTVS kick today. I did find one general criterion that implies that a locally convex space is paracompact. According to the Encyclopedia of Mathematics, if it is Montel (which means that it is barrelled and the Heine-Borel theorem holds true for it), then it is paracompact. Although this criterion is important, it is of no use to your specific question.

On the other hand, I have an answer to half of your specific question: I think that any direct sum of Frechet spaces of any cardinality is normal. One way (maybe the way) to understand normality is that it is equivalent to the Tietze extension theorem. If you know that a space is constructed from other spaces that are normal, you can try to use Tietze extension to confirm the normality axiom in the new space. If $A$ and $B$ are the two closed sets, you obtain disjoint open neighborhoods from a continuous function that is 0 on $A$ and 1 on $B$. Once you have confirmed the axiom, the powerful Tietze theorem gives you general Tietze extensions.

Let $X$ be a Frechet space. (Actually $X$ can be any pointed metrizable space. The marked point defines the coproduct operation of your question.) Let $\alpha$ be an ordinal, for instance the ordinal of a well-ordering of $\mathbb{R}$. Then $X^\alpha$ is normal by transfinite induction. Let $A$ and $B$ be disjoint closed sets in $X^\alpha$; we hope to extend the function that is $0$ on $A$ and $1$ on $B$. For $\beta < \alpha$, let $A_\beta$ and $B_\beta$ be the intersections of $A$ and $B$ with $X^\beta$. Suppose that $\alpha = \beta+1$ is a successor ordinal. Since $X^\beta$ is normal by induction, there is a Tietze extension of the function that is $0$ on $A_\beta$ and $1$ on $B_\beta$. It can be extended to $\beta+1$ because $X^\beta \cup A_{\beta+1} \cup B_{\beta+1}$, because that is a closed set in $X^{\beta+1}$. This establishes the extension for successor ordinals. Okay, this looks like a circular argument, but it's not. If $\beta$ is a finite ordinal, then $X^{\beta+1}$ is metrizable, so it is normal. If $\beta$ is an infinite ordinal, then actually $\beta$ and $\beta+1$ have the same cardinality, so we have already shown that $X^{\beta+1}$ is normal and we can use the Tietze theorem on it.

If instead $\alpha$ is a limit ordinal, then the extensions all the way up to $\alpha$ work just because they work. In order to understand the limit ordinal case, it is helpful to realize that the direct limit for direct sums in the category of locally convex spaces is just the box topology; in this special case it is the same as the direct limit in topological spaces. (Boxes happen to be convex.)

As for paracompactness, there is an interesting result called the Michael selection theorem which seems to do for paracompactness what the Tietze theorem does for normality. The question is whether there is a "Michael" version of the above argument. I would not be surprised if there is.