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Section 5.4 Locally Convex spaces

We met so far several spaces whose natural topology cannot be defined via a single norm but rather via a family of seminorms.
Definition 5.4.1. Seminorm.
A seminorm on a vector space \(X\) is a function \(p:X\to[0,\infty)\) such that
  1. \(p(x)\geq0\) for all \(x\in X\text{;}\)
  2. \(p(\lambda x)=|\lambda|p(x)\) for all \(\lambda\in\mathbb R\) and \(x\in X\text{;}\)
  3. \(p(x+y)\leq p(x)+p(y)\) for all \(x,y\in X\)
Since we are only interested in Hausdorff topologies, we use the following definition.
Definition 5.4.2. Locally Convex space.
By locally convex vector space we mean a vector space \(X\) whose topology is generated by a family of seminorms \(p_\alpha\text{,}\) \(\alpha\in I\text{,}\) such that, for every \(v\in X\text{,}\) there is some \(\alpha_0\in I\) such that \(p_{\alpha_0}(x)\neq0\text{.}\)
Definition 5.4.3. Cauchy sequences in locally convex spaces.
A sequence \(x_n\) is cauchy if, for every \(p_\alpha\) and \(\eps>0\text{,}\) there is a \(N>0\) such that \(p_\alpha(x_n-x)<\eps\) for every \(n\geq N\text{.}\)
The reader can verify the following facts:
  1. a sequence \(x_n\) converges to \(x\) in \(X\) if and only if \(p_\alpha(x_n-x)\to0\) for every \(\alpha\in I\text{;}\)
  2. every converging sequence is Cauchy.
Definition 5.4.4. Frechet space.
By Frechet space we mean a complete locally convex vector space \(X\) whose topology is generated by a countable family of seminorms \(p_n\text{.}\)
When the family is countable, this topology coincides with the topology induced by the translation-invariant distance function
\begin{equation*} d(x,y) = \sum_{n=1}^\infty \frac{1}{2^n}\frac{p_n(x-y)}{1+p_n(x-y)}. \end{equation*}
A space whose topology is induced by a distance function that makes it a complete metric space is called a completely metrizable space. Hence we can summarize the observation above by the following: We now go a bit in detail over the topology induced by a family of seminorms.

By topology induced by a family of seminorms \(p_\alpha\) we mean the coarsest (i.e. smallest) topology that makes all \(p_\alpha\) continuous.

Denote by \(\cal U_0\) the topology generated by the basic neighborhoods in the claim of the theorem. Clearly all \(p_\alpha\) are continuous under \(\cal U_0\) since, for every \(\alpha\in I\text{,}\) the set \(\{p_\alpha}(x)\leq\eps\}\) is a basic neighborhood of zero.

Now, let \(\cal U\) be any other topology for which all \(p_\alpha\) are continuous. Then, since finite intersections of open sets are open, for any finite choice of indices \(\alpha_1,\dots\alpha_k\text{,}\)
\begin{equation*} \{p_{\alpha_1}(x)\leq\eps\}\cap\dots\cap\{p_{\alpha_k}(x)\leq\eps\}\subset\cal U \end{equation*}
Hence, \(\cal U_0\subset\cal U\text{.}\)
There is an important difference between locally convex spaces with countable family of seminorms and whose with uncountably many. To understand this difference, we go over a particular case.

Endow \(C^0(\bS^1)\) with the family of seminorms
\begin{equation*} p_x(f) = |f(x)|,\;x\in\bS^1. \end{equation*}
This family is uncountable and convergence in this topology means pointwise convergence:
\begin{equation*} f_n\to f\text{ if and only if }f_n(x)\to f(x)\text{ for every }x\in\bS^1. \end{equation*}
Now, consider the set
\begin{equation*} A = \{f\in C^0(\bS^1)\,:\, f(q)=0\text{ for infinitely many }q\in\bQ\}. \end{equation*}
1. \(A\) is sequencially closed. Indeed, assume that \(f_n\to f\) and that \(f\) has only finitely many zeros in \(\bQ\text{.}\) Denote by \(Z\) be the set of these zeros. Since \(f_n\to f\text{,}\) for each \(q\in\bQ\setminus Z\text{,}\) there is a \(N(q)\) such that \(f_n(q)\neq0\) for all \(n\geq N(q)\text{.}\) This, though it is not possible since, by definition, given any \(f_n\text{,}\) there is some \(q\in\bQ\setminus Z\) such that \(f_n(q)=0\text{.}\) Hence, \(f\) must have infinitely many zeros in \(\bQ\text{,}\) namely \(f\in A\text{.}\)

2. \(A\) is not closed. Take \(f(x)=1\text{.}\) By the argument above, \(f\) cannot be the sequence of elements of \(A\text{.}\) Yet, \(f\) lies in the closure of \(A\text{.}\) Indeed, basic neighborhoods of \(f\) are given by sets
\begin{equation*} U_{x_1,\dots,x_k}(\eps) = \{g\in C^0(\bS^1) \,:\, |g(x_i)-1|<\eps\text{ for }i=1,\dots,k\}. \end{equation*}
Let \(W=\{x_1,\dots,x_k\}\) and \(x\in\bS^1\setminus W\text{,}\) choose a sequence \(q_n\to x\text{,}\) \(q_n\in\bQ\) and set
\begin{equation*} Z = \{x\}\cup\{q_1,q_2,\dots\}. \end{equation*}
Since \(W\) and \(Z\) are closed and disjoint, by Urysohn's Lemma there exist a continuous function \(g:\bS^1\to[0,1]\) such that \(g|_Z=0\) and \(g|_W=1\text{.}\) Then \(g\in A\) and \(g\in U_{x_1,\dots,x_k}(\eps)\) for all \(\eps>0\text{.}\) Hence, \(f\) lies in the closure of \(A\text{.}\)

This example shows clearly that sequences are not able to fully encode the topology induced by an uncountable family of seminorms. A tool that can be used in these cases is defined below.
Definition 5.4.7.
Directed set A directed set is a non-empty set \(A\) together with a preorder, namely a binary relation \(\succ\) such that, for each \(a,b\in A\text{,}\) there is a \(c\in A\) such that \(c\succ a\) and \(c\succ b\text{.}\)
For instance, \(\bN,\bZ,\bQ,\bR\) are all directed sets with respect to \(\geq\text{.}\) \(\bC\) is a directed set, for instance, with respect to \(c_1=x_1+iy_1\prec c_2=x_2+iy_2\) if \(x_1\geq x_2\text{.}\)
Definition 5.4.8.
Net A net on a set \(X\) is a map \(n:A\to X\) whose domain \(A\) is a direct set.
In other words, nets are sequences with steroids: the set of indices in a net can be uncountable.
Definition 5.4.9.
Convergence of a net We say that a net \(x_\alpha\text{,}\) \(\alpha\in I\text{,}\) in \(X\) converges to a point \(x\in X\) if, for every neighborhood \(U\) of \(x\text{,}\) there is an \(a\in I\) such that \(x_b\in U\) for every \(b\succ a\text{.}\)
We now complete the example above about \(C^0(\bS^1)\) with the following observation.

There is a net in \(A\) converging to \(f(x)=1\text{.}\) Let \(I\) be the collection of all finite subsets of \(\bS^1\text{.}\) This is a directed set with respect to inclusion: given any finite subsets \(F\) and \(G\text{,}\) define \(H=F\cup G\text{.}\) Then \(H\supset F\) and \(H\supset G\text{.}\) Now, fix some \(r\in\bS^1\setminus\bQ\text{.}\) Given any \(F\in I\text{,}\) choose any infinite sequence \(q_n\in\bQ\setminus F\) converging to \(r\) and set
\begin{equation*} Z_F = \{r\}\cup\{q_1,q_2,\dots\}. \end{equation*}
Since \(F\) and \(Z_F\) are closed and disjoint, there is a \(g_F\in C^0(\bS^1)\) such that
\begin{equation*} g_F|_F=1\text{ and }g_F|_{Z_F}=0. \end{equation*}
Hence, \(g_F, F\in I\) is a net in \(A\text{.}\) We claim that \(g_F\to f\text{.}\) Indeed, given any \(x_0\in\bS^1\text{,}\) set \(F_0=\{x_0\}\in I\text{.}\) Then, for every \(F\succ F_0\) (i.e. for every \(F\subset\bS^1\) such that \(x_0\in F\)), we have that \(g_F(x_0)=1\) (since \(g_F|_F=1\)). Hence, let \(U\) be a basic neighborhood of \(f\text{,}\) namely
\begin{equation*} U = \{g\in C^0(\bS^1) \,:\, |g(x_i)-1|<\eps\text{ for }i=1,\dots,k\}. \end{equation*}
for some points \(x_1,\dots,x_k\) and some \(\eps>0\text{,}\) and set \(G=\{x_1,\dots,x_k\}\text{.}\) Then, for every \(F\succ G\text{,}\) we have that \(g_F(x_i)=1\) for every \(i=1,\dots,k\text{,}\) so that \(g_F\in U\text{.}\) This means precisely that \(g_F\to f\text{.}\)

This long example on pointwise convergence topology suggests that nets plays precisely the role of sequences even when the topology is generated by uncountably many seminorms. In fact, we have the following results, of which we omit the proofs.