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I'm confused by a set of lecture notes I'm reading and would like help in understanding what's going on. First, there is the following nice theorem.

Theorem. The topology of a locally convex space is the limit topology w.r.t the collection of seminorms obtained as Minkowski functionals associated to a local basis of $0$ consisting of convex, balanced, open sets.

Then the notes go on to define $C^\infty _c(U)$ as a colimit, with a colimit topology:

$C^\infty _c(U)$ is defined to be the colimit in $\mathsf{Top}$ $$C_c^\infty (U)=\varinjlim _{K\subset U}C^\infty (K),\;\;\;\;K\text{ compact}$$ The topology on this colimit is the colimit topology w.r.t to the following family of seminorms $$p_{K,n}(f)=\sup \left\{|\partial ^\alpha f(x)|:x\in K,|\alpha|\leq n \right\}.$$

What bugs me is the "lack of uniformity" here, since we sometimes topologize with the limit topology w.r.t the family of seminorms and sometimes with the colimit topology.

Isn't there some uniform approach to topologizing these spaces?

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    $\begingroup$ If you declare an absolutely convex set $V \subseteq C_c^\infty(U)$ a $0$-neighbourhood if, for all $K$ there are $n\in\mathbb N$ and $\varepsilon$ such that $V\cap C^\infty(K) \supset \lbrace f\in C^\infty(K): |\partial^\alpha f(x) |\le \varepsilon$ for all $|\alpha|\le n\rbrace$ -- then the (locally convex) colimit topology is the limit topology. $\endgroup$ Commented Jul 28, 2015 at 10:43
  • $\begingroup$ @Jochen I don't understand your comment. Can you post a more detailed version as an answer? $\endgroup$ Commented Jul 28, 2015 at 11:19

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In Functional Analysis, one often speaks of projective limits (limits in the category LCS of locally convex spaces) and inductive limits (colimits in LCS). In the latter case one has to be careful as colimits in TOP usually differ from those in LCS. What you state as a theorem says that every locally convex space is a projective limit of seminormed spaces.

To apply this little theorem to $C_c^\infty(U)$ one needs a locally convex topology and in my comment I gave a description of a system of $0$-neighbourhoods of such a topology, explicitely a subset $A$ of $C_c^\infty(U)$ is open if, for every $g\in A$ the set $A-g$ contains an absolutely convex set $V$ such that, for every compact $K\subseteq U$ there are $n\in\mathbb N$ and $\varepsilon>0$ such that $V\supset \lbrace f\in C^\infty(K): |\partial^\alpha f(x)|\le \varepsilon$ for all $|\alpha|\le n\rbrace$. This is not very handy for concrete calculations but it is quite easy to see that the topology satisfies the universal property a colimit should have: A linear map $C_c^\infty(U) \to X$ (where $X$ is any locally convex space) is continuous if and only if all restrictions $T|_{C^\infty(K)}$ are continuous.

Contrary to that theorem, it is not true that every locally convex space is an inductive limit of seminormed spaces. Such space are called bornological.

There is also a caegory BOR of vector spaces endowed with a so-called bornology (a system of bounded sets). In this category, colimits are much easier but there is a high price to pay: the Hahn-Banach theorem is no longer true in BOR.

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  • $\begingroup$ Last sentence of the first paragraph - in which category is the projective limit taken? Also, I'm still confused. It seems then that the topology on $C^\infty _c(U)$ is the colimit topology w.r.t inclusions, and is also the limit topology w.r.t the seminorms $p_{K,n}$, contrary to what I wrote in the body of the question. Is this correct? $\endgroup$ Commented Aug 1, 2015 at 14:17
  • $\begingroup$ Systematically, the (projective) limit should be taken in LCS. However, it coincides with the limit in TOP. And you are right: the colimit topology of $C_c^\infty(U)$ equals the limit topology w.r.t. $P_{K,n}$. $\endgroup$ Commented Aug 3, 2015 at 6:24
  • $\begingroup$ What kind of (co)limits does the forgetful functor $\mathsf{LCS}\rightarrow \mathsf{Top}$ reflect? $\endgroup$ Commented Aug 3, 2015 at 9:25

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