> [!definition] > > Let $E$ be a [[Vector Space|vector space]] over $K \in \bracs{\real, \complex}$ and $\seqi{E}$ with $\bracsn{T_j^i: i, j \in I, i \lesssim j}$ be a [[Directed Set|directed system]] of [[Locally Convex Topological Vector Space|locally convex topological vector spaces]], then there exists a locally convex topological vector space $E$ and a family of continuous linear maps $\bracsn{T^i_E: i \in I}$ such that > 1. $T^i_E: E_i \to E$ is a continuous linear map. > 2. $T^i_j \circ T^j_E = T^i_E$ for all $i, j \in I$. > 3. If $F$ is another locally convex topological vector space, and $T: E \to F$ is a linear map, then $T$ is continuous if and only if $T \circ T^i_E$ is continuous for each $i \in I$. > 4. Let $[\cdot]: E \to \real^+$ be a [[Seminorm|seminorm]] on $E$, then $[\cdot]$ is continuous if and only if $[\cdot] \circ T^i_E$ is continuous for all $i \in I$. > 5. If $(F, \bracsn{T_F^i: i \in I})$ is another pair satisfying $(1)$ and $(2)$, then there exists a unique continuous linear map $T^E_F: E \to F$ such that $T^i_F = T^E_F \circ T^i_E$ for all $i \in I$. > > For each $i \in I$, let $N_i$ be a [[Neighbourhood Base|neighbourhood base]] at $0$ for $E_i$ consisting of [[Convexity|convex]], [[Balanced Set|balanced]], and [[Absorbing|absorbing]] sets, and $N$ be the family of all convex, balanced, and absorbing sets in $E$. Let > $ > N_I = \bracs{U \in N: (T^i_{E})^{-1}(U) \in N_i \forall i \in I} > $ > be the family of convex, balanced, and absorbing sets that get pulled back into open sets in each $E_i$. For each $U \subset E$, let $\Gamma(U)$ be its convex hull, and let > $ > N_C = \bracs{\Gamma\paren{\bigcup_{i \in I}T^i_E(U_i)}: U_i \in N_i} > $ > be the convex closure of each combination of $\bracs{N_i: i \in I}$. Then, > > 6. Both $N_I$ and $N_C$ form a neighbourhood base at $0$ for $E$. > > In other words, [[Direct Limit|direct limits]] exist in the category of locally convex topological vector spaces over $K$, with its open sets characterised from the convex, balanced, and absorbing sets from each $E_i$. > > *Proof*. $(1)$, $(2)$, $(4)$, existence: By the existence of the limit in the category of $K$-vector spaces. > > $(1)$, continuity: Let $\mathscr{T} = \seqi{\mathcal T}$ be the collection of all locally convex topologies on $E$ such that $T^i_E: E_i \to E$ is continuous for each $i \in I$. Note that $\mathscr{T}$ is non-empty as the trivial topology satisfies the given criteria. Let $\mathcal T$ be the [[Initial Topology on Topological Vector Space|initial topology]] generated by the identity $I: E \to (E, \mathcal T_i)$ into each topology, then $\mathcal T$ is a topology on $E$ with $T^i_E: E_i \to (E, \mathcal T)$ being continuous for all $i \in I$. > > $(3)$: Let $F$ be a locally convex topological vector space and $T: E \to F$ be a linear map such that $T \circ T^i_E$ is continuous for each $i \in I$, then the topology on $E$ generated by $F$ is locally convex, and belongs to $\mathscr{T}$. Therefore $T$ is continuous. > > $(4)$, $(5)$: Follows directly from $(3)$. > > $(6)$: The vector space topology generated by $N_I$ and $N_C$ are both locally convex, with each $T^i_E: E_i \to E$ being continuous, so topology on $E$ contains $N_I$ and $N_C$. > > Let $U \subset E$ be a convex, balanced, and absorbing neighbourhood of $0$, then $(T^i_E)^{-1}(U)$ is a convex, balanced, and absorbing neighbourhood of $0$ in $E_i$ for each $i \in I$. Therefore $U$ is in the vector space topology generated by $N_I$. In addition, $U$ contains the convex hull generated by $U \cap T^i_E(E_i)$ for all $i \in I$, which is an element of $N_C$. Therefore $N_I$ and $N_C$ are neighbourhood bases at $0$ for the limit topology.