8.9 Projective Limits
Definition 8.9.1 (Projective Uniformity). Let $E$ be a vector space over $K \in \RC$, $\seqi{F}$ be TVSs over $K$, and $\seqi{T}$ where $T_{i} \in \hom(E; F_{i})$ for all $i \in I$, then there exists a uniformity $\fU$ on $E$ such that:
For each $i \in I$, $T_{i} \in L(E; F_{i})$.
If $\mathfrak{V}$ is a uniformity on $E$ satisfying $(1)$, then $\mathfrak{V}\supset \fU$.
Moreover,
$\fU$ is translation-invariant.
$E$ equipped with the topology induced by $\fU$ is a topological vector space.
For any TVS $F$ over $K$ and linear map $T \in \hom(F; E)$, $T \in L(F; E)$ if and only if $T_{i} \circ T \in L(F; F_{i})$ for all $i \in I$.
The collection
\[\bracs{\bigcap_{j \in J}T_j^{-1}(U_j) \bigg | J \subset I \text{ finite}, U_j \in \cn_{F_j}(0)}\]is a fundamental system of neighbourhoods for $E$ at $0$.
The uniformity $\fU$ and its topology are the projective uniformity/topology induced by $\seqi{T}$.
Proof. (1), (U): By Definition 5.2.3.
Let $U \in \fU$, then there exists $J \subset I$ finite and translation-invariant entourages $\seqj{U}$ such that
(3): For each $j \in J$, $(x, y) \in (T_{j} \times T_{j})^{-1}(U_{j})$, and $z \in E$,
so $(T_{j} \times T_{j})^{-1}(U_{j})$ is translation-invariant, and so is $V$.
(4): By (TVS1) and (TVS2), for each $j \in J$, there exists an entourage $V_{j}$ of $F_{j}$ and $\eps_{j} > 0$ such that for any $(x, x'), (y, y') \in V_{j}$ and $\lambda, \lambda' \in K$ with $\abs{\lambda - \lambda'}< \eps_{j}$, $(x + y, x' + y'), (\lambda x, \lambda' x') \in U_{j}$.
Therefore, for any $(x, x'), (y, y') \in \bigcap_{j \in J}T_{j}^{-1}(V_{j})$ and $\lambda, \lambda' \in K$ with $\abs{\lambda - \lambda'}< \min_{j \in J}\eps$, $(x + y, x' + y'), (\lambda x, \lambda' x') \in V$.
(5): By Definition 8.5.1 and (4) of Definition 5.2.3.
(6): By Definition 5.2.3.$\square$
Definition 8.9.2 (Projective Limit of Topological Vector Spaces). Let $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$ be a downward-directed system of topological vector spaces over $K \in \RC$, then there exists $(E, \bracsn{T^E_i}_{i \in I})$ such that:
$E$ is a TVS over $K$.
For each $i \in I$, $T^{E}_{i} \in L(E; E_{i})$.
For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
\[\xymatrix{ E_i \ar@{->}[r]^{T^i_j} & E_j \\ E \ar@{->}[u]^{T^E_i} \ar@{->}[ru]_{T^E_j} & }\]For any pair $(F, \bracsn{S^F_i}_{i \in I})$ satisfying (1), (2), and (3), there exists a unique $S \in L(F; E)$ such that the following diagram commutes
\[\xymatrix{ & E_i \\ F \ar@{->}[r]_{S} \ar@{->}[ru]^{S^F_i} & A \ar@{->}[u]_{T^E_i} }\]for all $i \in I$.
For any TVS $F$ over $K$ and $S \in \hom(F; E)$, $S \in L(F; E)$ if and only if $T^{E}_{i} \circ S \in L(F; E_{i})$ for all $i \in I$.
The pair $(E, \bracsn{T^E_i}_{i \in I})$ is the projective limit of $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$.
Proof. Let $(E, \bracsn{T^E_i}_{i \in I})$ be the inverse limit of $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$ as $K$-vector spaces (Proposition 1.2.10).
Equip $E$ with the projective topology generated by $\bracsn{T^E_i}_{i \in I}$, then $(E, \bracsn{T^E_i}_{i \in I})$ satisfies (1), (2), and (3).
(5): By (5) of Definition 8.9.1.
(U): By (U) of Proposition 1.2.10, there exists a unique $S \in \hom(F; E)$ such that the given diagram commutes. By (4), $S \in L(F; E)$.$\square$