\section{Projective Limits}
\label{section:tvs-projective-limit}

\begin{definition}[Projective Uniformity]
\label{definition:tvs-initial}
    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:
    \begin{enumerate}
        \item For each $i \in I$, $T_i \in L(E; F_i)$.
        \item[(U)] If $\mathfrak{V}$ is a uniformity on $E$ satisfying $(1)$, then $\mathfrak{V} \supset \fU$.
    \end{enumerate}

    Moreover,
    \begin{enumerate}
        \item[(3)] $\fU$ is translation-invariant.
        \item[(4)] $E$ equipped with the topology induced by $\fU$ is a topological vector space.
        \item[(5)] 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 $i \in I$.
        \item[(6)] 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 $0$.
    \end{enumerate}

    
    The uniformity $\fU$ and its topology are the \textbf{projective uniformity/topology} induced by $\seqi{T}$.
\end{definition}
\begin{proof}
    (1), (U): By \autoref{definition:initial-uniformity}.

    Let $U \in \fU$, then there exists $J \subset I$ finite and translation-invariant entourages $\seqj{U}$ such that
    \[
    U \subset V = \bigcap_{j \in J}(T_j \times T_j)^{-1}(U_j)
    \]


    (3): For each $j \in J$, $(x, y) \in (T_j \times T_j)^{-1}(U_j)$, and $z \in E$,
    \[
    (T_j \times T_j)(x + z, y + z) = (T_jx + T_jz, T_jy + T_jz) \in U_j
    \]

    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 \autoref{definition:continuous-linear} and (4) of \autoref{definition:initial-uniformity}.

    (6): By \autoref{definition:initial-uniformity}.
\end{proof}

\begin{definition}[Projective Limit of Topological Vector Spaces]
\label{definition:tvs-projective-limit}
    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:
    \begin{enumerate}
        \item $E$ is a TVS over $K$.
        \item For each $i \in I$, $T^E_i \in L(E; E_i)$.
        \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
        \[
        \xymatrix{
        E_i \ar@{->}[r]^{T^i_j} & E_j \\
        E \ar@{->}[u]^{T^E_i} \ar@{->}[ru]_{T^E_j} &
        }
        \]

        \item[(U)] 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 $i \in I$.
        \item 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 $i \in I$.
    \end{enumerate}

    The pair $(E, \bracsn{T^E_i}_{i \in I})$ is the \textbf{projective limit} of $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$.
\end{definition}
\begin{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 (\autoref{proposition:module-inverse-limit}).

    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 \autoref{definition:tvs-initial}.

    (U): By (U) of \autoref{proposition:module-inverse-limit}, there exists a unique $S \in \hom(F; E)$ such that the given diagram commutes. By (4), $S \in L(F; E)$.
\end{proof}
