9.4 Projective Limits
Proposition 9.4.1. Let $E$ be a vector space over $K \in \RC$, $\seqi{F}$ be locally convex spaces over $K$, and $\seqi{T}$ where $T_{i} \in \hom(E; F_{i})$ for all $i \in I$, then the projective topology on $E$ is locally convex.
Proof. By Definition 8.9.1,
is a fundamental system of neighbourhoods at $0$. For each $i \in I$, $U_{i} \in \cn_{F_i}(0)$ convex, $T^{-1}(U_{i})$ is also convex. Since each $F_{i}$ is locally convex, $\mathcal{B}$ contains a fundamental system of neighbourhoods at $0$ consisting of only convex sets.$\square$
Proposition 9.4.2. Let $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$ be a downward-directed system of locally convex spaces over $K \in \RC$, then $E = \lim_{\longleftarrow}E_{i}$ is locally convex.
Proof. By (U) of Definition 8.9.2 and Definition 8.9.1, $E$ is equipped with the projective topology generated by the projection maps $E \to E_{i}$. By Proposition 9.4.1, $E$ is locally convex.$\square$
Proposition 9.4.3. Let $E$ be a Hausdorff complete locally convex space over $K \in \RC$, $\mathcal{B}\subset \cn_{E}(0)$ be a fundamental system of neighbourhoods consisting of convex, circled, and radial sets, directed under inclusion.
For each $U \in \mathcal{B}$, let $[\cdot]_{U}$ be its gauge, $M_{U} = \bracs{x \in E|[\cdot]_U = 0}$, $E_{U} = E/M_{U}$, and $\norm{\cdot}_{U}: E_{U} \to [0, \infty)$ be the quotient of $[\cdot]_{U}$ by $M_{U}$, then
For each $U, V \in \mathcal{B}$ with $U \subset V$, let
\[\pi^{U}_{V}: E_{U} \to E_{V} \quad x + M_{U} \mapsto x + M_{V}\]then $\pi^{U}_{V} \in L(E_{U}; E_{V})$.
$(\bracsn{E_U}_{U \in \mathcal{B}}, \bracs{\pi^U_V|U, V \in \mathcal{B}, U \subset V})$ is a downward-directed system of topological vector spaces.
The map $\pi \in L(E, \lim_{\longleftarrow}E_{U})$ induced by $\bracs{\pi_U}_{U \in \mathcal{B}}$ is a bijection.
For each $U, V \in \mathcal{B}$, let $\ol E_{U}$ be the completion of $E_{U}$, $\ol{\pi_U}\in L(E; \ol E_{U})$, and $\ol{\pi^U_V}\in L(\ol E_{U}; \ol E_{V})$ be the unique extensions of $\pi_{U}$ and $\pi^{U}_{V}$, respectively. Then,
\[E = \lim_{\longleftarrow}E_{U} = \lim_{\longleftarrow}\ol E_{U}\]
Proof. (1): Since $V \supset U$, $[\cdot]_{V} \ge [\cdot]_{U}$, so $M_{V} \supset M_{U}$. Thus $\ker(\pi_{V}) \supset M_{U}$. By (U) of the quotient, $\pi_{V}$ factors through $E_{U}$ as $\pi^{U}_{V}$, so $\pi^{U}_{V} \in L(E_{U}; E_{V})$.
(2): Since $\mathcal{B}$ is a fundamental system of neighbourhoods, it is downward-directed under inclusion. For any $U, V, W \in \mathcal{B}$ with $U \subset V \subset W$, $M_{U} \supset M_{V} \supset M_{W}$. Thus $\pi^{U}_{W} = \pi^{V}_{W} \circ \pi^{U}_{V}$.
(3): Let $\lim E_{U}$ be the projective limit. For each $U \in \mathcal{B}$, let $p_{U}: \lim E_{U} \to E_{U}$ be the canonical map.
Let $x \in E$. Since $E$ is Hausdorff and $\mathcal{B}$ is a fundamental system of neighbourhoods at $0$, there exists $U \in \mathcal{B}$ such that $\pi_{U}(x) \ne 0$. In which case, $p_{U} \circ \pi(x) = \pi_{U}(x) \ne 0$, so $\pi$ is injective.
Let $x \in \lim E_{U}$. For each $U \in \mathcal{B}$, let $x_{U} \in E$ such that $\pi_{U}(x_{U}) = p_{U}(x)$. For any $V \in \cn_{E}(0)$, there exists $W \in \mathcal{B}$ with $W \subset V$. In which case, for any $U \in \mathcal{B}$ with $U \subset W$,
Thus for any $U' \in \mathcal{B}$ with $U \subset W$, $[x_{U} - x_{U'}]_{W} = 0$, and $x_{U} - x_{U'}\in W$. Therefore $\bracs{x_U}_{U \in \mathcal{B}}$ is a Cauchy net, and converges to $x_{0} \in E$ by completeness of $E$.
For any $U \in \mathcal{B}$, $\pi_{U}(x_{0}) = \lim_{V \in \mathcal{B}}\pi_{U}(x_{V}) = p_{U}(x)$, so $\pi(x_{0}) = x$, and $\pi$ is surjective.
(4): Since $\mathcal{B}\subset \cn_{E}(0)$ is a fundamental system of neighbourhoods, the topology on $E$ is the projective topology generated by $\bracs{\pi_U|U \in \mathcal{B}}$. As $\pi_{U} \circ \pi^{-1}= p_{U} \in L(\lim E_{U}; E_{U})$ for all $U \in \mathcal{B}$, $\pi^{-1}\in L(\lim E_{U}; E)$ by (U) of the projective topology.
Let $x \in \lim\ol{E}_{U}$ and $V \in \cn(x)$. Since $\mathcal{B}$ is downward-directed and $\lim\ol{E}_{U}$ is equipped with the projective topology induced by $\bracs{p_U|U \in \mathcal{B}}$, there exists $U \in \mathcal{B}$ and $W \in \cn_{\ol E_U}(x)$ such that $p_{U}^{-1}(W) \subset V$. As $E_{U}$ is dense in $\ol E_{U}$, there exists $y_{U} \in W$, and $y \in E$ such that $y_{U} = \pi_{U}(y)$. Therefore $\pi(y_{U}) \in p_{U}^{-1}(W) \subset V$, and $\lim E_{U}$ is dense in $\lim \ol{E}_{U}$.
Since $E$ is complete and isomorphic to $\lim E_{U}$, $\lim E_{U}$ is a complete, and thus closed subset of $\lim \ol{E}_{U}$. Therefore $E = \lim E_{U} = \lim \ol{E}_{U}$.$\square$