\section{Inductive Limits}
\label{section:tvs-inductive}

\begin{definition}[Inductive Topology]
\label{definition:tvs-inductive}
    Let $\seqi{E}$ be TVSs over $K \in \RC$, $\seqi{T}$ such that $T_i \in \hom(E_i; E)$ for all $i \in I$, and $E$ be a vector space over $K$, then there exists a topology $\topo$ on $E$ such that:
    \begin{enumerate}
        \item $(E, \topo)$ is a TVS over $K$.
        \item For each $i \in I$, $T_i \in L(E_i; E)$.
        \item[(U)] For any topology $\mathcal{S}$ on $E$ satisfying (1) and (2), $\mathcal{S} \subset T$.
        \item For any TVS $F$ and $T \in \hom(E; F)$, $T \in L(E; F)$ if and only if $T \circ T_i \in L(E_i; F)$ for all $i \in I$.
    \end{enumerate}

    The topology $\topo$ is the \textbf{inductive topology} on $E$ induced by $\seqi{T}$.
\end{definition}
\begin{proof}
    (1): Let
    \[
    \mathcal{B} = \bracs{U \subset E|U \text{ radial, circled}, T_i^{-1}(U) \in \cn_{E_i}(0) \forall i \in I}
    \]

    To see that $\mathcal{B}$ is a fundamental system of neighbourhoods at $0$ for a vector space topology on $E$, it is sufficient to verify the following and apply \autoref{proposition:tvs-0-neighbourhood-base}.
    \begin{enumerate}
        \item[(TVB1)] Every set in $\mathcal{B}$ is radial and circled by definition.
        \item[(TVB2)] For any $U \in \mathcal{B}$, $U$ is circled, so $\frac{1}{2}U + \frac{1}{2}U \subset U$. Since $\frac{1}{2}U$ is also circled and radial, $\frac{1}{2}U \in \mathcal{B}$.
    \end{enumerate}

    Let $\topo$ be the vector space topology such that $\mathcal{B}$ is a fundamental system of neighbourhoods at $0$, then $(E, \topo)$ is a TVS.

    (2): For each $i \in I$ and $U \in \mathcal{B}$, $T_i^{-1}(U) \in \cn_{E_i}(0)$, so $T_i \in L(E_i; E)$.

    (U): Let $U \in \cn_{(E, \mathcal{S})}(0)$ be circled and radial, then by (2), $T_i^{-1}(U) \in \cn_{E_i}(0)$ for all $i \in I$. Thus the circled and radial neighbourhoods of $0$ in $(E, \mathcal{S})$ is a subset of $\mathcal{B}$.

    (4): Let $U \in \cn_F(0)$ be circled and radial and $i \in I$. Since $T \circ E_i \in L(E_i; F)$, $T_i^{-1}(T^{-1}(U)) \in \cn_{E_i}(0)$, so $T^{-1}(U) \in \mathcal{B} \subset \cn_E(0)$.
\end{proof}

\begin{definition}[Direct Sum]
\label{definition:tvs-direct-sum}
    Let $\seqi{E}$ be TVSs over $K \in \RC$, then there exists $(E, \seqi{\iota})$ such that:
    \begin{enumerate}
        \item $E$ is a TVS over $K$. 
        \item For each $i \in I$, $\iota_i \in L(E_i; E)$. 
        \item[(U)] For each $(F, \seqi{T})$ satisfying (1) and (2), there exists a unique $T \in L(E; F)$ such that the following diagram commutes:
        \[
        \xymatrix{
        E \ar@{->}[r]^{T} & F \\
        E_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & 
        }
        \]
    \end{enumerate}

    
    The space $E = \bigoplus_{i \in I}E_i$ is the \textbf{direct sum} of $\seqi{E}$. 
\end{definition}
\begin{proof}
    Let $(E, \seqi{\iota})$ be the direct sum of $\seqi{E}$ as vector spaces, and equip it with the inductive topology induced by $\seqi{\iota}$, then $(E, \seqi{\iota})$ satisfies (1) and (2). 

    (U): By (U) of the \hyperref[direct sum]{definition:direct-sum}, there exists a unique $T \in \hom(E; F)$ such that the diagram commutes. In which case, by (4) of \autoref{definition:tvs-inductive}, $T \in L(E; F)$. 
\end{proof}

\begin{proposition}
\label{proposition:finite-tvs-product}
    Let $\seqf{E_j}$ be TVSs over $K \in \RC$, then
    \[
    \prod_{j = 1}^n E_j = \bigoplus_{j = 1}^n E_j
    \]
\end{proposition}
\begin{proof}
    Let $1 \le k \le n$, then for each $1 \le k, l \le n$, $\pi_l \circ \iota_k \in L(E_k, E_l)$, so by (U) of the \hyperref[product]{definition:tvs-product}, $\iota_k \in L(E_k; \prod_{j = 1}^n E_j)$. Thus $\prod_{j = 1}^n E_j$ satisfies (1) and (2) of the \hyperref[direct sum]{definition:tvs-direct-sum}. 

    For any TVS $F$ over $K$ and $\seqf{T_j}$ with $T_j \in L(E_j; F)$ for each $1 \le j \le n$, let
    \[
    T: \prod_{j = 1}^n E_j \to F \quad (x_1, \cdots, x_n) \mapsto \sum_{j = 1}^n T_jx_j
    \]

    then $T \in L(\prod_{j = 1}^n E_j; F)$ is the unique continuous linear map such that the following diagram commutes:
    \[
    \xymatrix{
    E \ar@{->}[r]^{T} & F \\
    E_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & 
    }
    \]

    Hence $\prod_{j = 1}^n E_j$ satisfies (U) of the \hyperref[direct sum]{definition:tvs-direct-sum}, so the spaces coincide. 
\end{proof}



\begin{definition}[Inductive Limit]
\label{definition:tvs-inductive-limit}
    Let $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of TVSs over $K \in \RC$, then there exists $(E, \bracsn{T^i_E}_{i \in I})$ such that:
    \begin{enumerate}
        \item $E$ is a TVS over $K$.
        \item For each $i \in I$, $T^i_E \in L({E_i, E})$.
        \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:

        \[
        \xymatrix{
        E_i \ar@{->}[rd]_{T^i_E} \ar@{->}[r]^{T^i_j} & E_j \ar@{->}[d]^{T^j_E} \\
        & E
        }
        \]


        \item[(U)] For any pair $(F, \bracsn{S^i_F}_{i \in I})$ satisfying (1), (2), and (3), there exists a unique $S \in L({E, F})$ such that the following diagram commutes

        \[
        \xymatrix{
        E_i \ar@{->}[d]_{T^i_E} \ar@{->}[rd]^{S^i_F} &  \\
        E \ar@{->}[r]_{S} & F
        }
        \]


        for all $i \in I$.
        \item For any TVS $F$ and $T \in \hom(E; F)$, $T \in L(E; F)$ if and only if $T \circ T^i_E \in L(E_i; F)$ for all $i \in I$.
    \end{enumerate}

    The pair $(E, \bracsn{T^i_E}_{i \in I})$ is the \textbf{inductive limit} of $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$.
\end{definition}
\begin{proof}
    Let $(E, \bracsn{T^i_E}_{i \in I})$ be the direct limit of $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ as vector spaces over $K$ (\autoref{proposition:module-direct-limit}). Equip $E$ with the \hyperref[inductive topology]{definition:tvs-inductive} induced by $\bracsn{T^i_E}_{i \in I}$, then $(E, \bracsn{T^i_E}_{i \in I})$ satisfies (1), (2), and (3).

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

    (5): By (5) of \autoref{definition:tvs-inductive}.
\end{proof}

\begin{definition}[Strict]
\label{definition:lc-inductive-strict}
    Let $(\seqi{E}, \bracsn{\iota^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of TVSs over $K \in \RC$, then the system is \textbf{strict} if:
    \begin{enumerate}
        \item For each $i, j \in I$ with $i \lesssim j$, $\iota^i_j: E_i \to E_j$ is injective.
        \item For each $i, j \in I$ with $i \lesssim j$, the topology of $E_i$ is induced by $\iota^i_j$.
    \end{enumerate}
\end{definition}