Added path lemma.

src/fa/lc/convex.tex

@@ -186,7 +186,7 @@
 
 \begin{definition}[Locally Convex Space]
 \label{definition:locally-convex}
-    Let $E$ be a TVS over $\RC$, then the following are equivalent:
+    Let $E$ be a TVS over $K \in \RC$, then the following are equivalent:
     \begin{enumerate}
         \item There exists a fundamental system of neighborhoods at $0$ consisting of convex sets.
         \item There exists a fundamental system of neighbourhoods at $0$ consisting of convex, circled, and radial sets.
@@ -203,6 +203,10 @@
     $(3) \Rightarrow (1)$: For each $i \in I$ and $r > 0$, $\bracs{x \in E| [x]_i < r}$ is convex.
 \end{proof}
 
+\begin{definition}[Fréchet Space]
+\label{definition:frechet-space}
+    Let $E$ be a locally convex space over $K \in \RC$, then $E$ is a \textbf{Fréchet space} if $E$ is first countable and complete. 
+\end{definition}
 
 
 \begin{definition}[Associated Normed Space]

src/fa/lc/inductive.tex

@@ -268,3 +268,16 @@
 
     which contradicts the fact that $(F_n + U) \cap E_n = \emptyset$.
 \end{proof}
+
+\begin{definition}[Space of Type (LB)]
+\label{definition:lb-space}
+    Let $E$ be a locally convex space, then $E$ is of type \textbf{(LB)} if $E$ is the strict inductive limit of a countable system of Banach spaces. 
+\end{definition}
+
+\begin{definition}[Space of Type (LF)]
+\label{definition:lf-space}
+    Let $E$ be a locally convex space, then $E$ is of type \textbf{(LF)} if $E$ is the strict inductive limit of a countable system of Fréchet spaces. 
+\end{definition}
+
+
+