Symmetry of the derivative (normed/frechet).

src/fa/tvs/continuous.tex

@@ -34,43 +34,6 @@
     Let $U \in \cn_F(0)$, then $T^{-1}(U) \in \cn_E(0)$, so there exists $\lambda \in K$ such that $\lambda T^{-1}(U) = T^{-1}(\lambda U) \supset B$ and $\lambda U \supset T(B)$.
 \end{proof}
 
-
-\begin{definition}[Initial Uniformity]
-\label{definition:tvs-initial}
-    Let $E$ be a vector space over $K \in \RC$, $\seqi{F}$ be TVSs, 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 $i \in I$.
-    \end{enumerate}
-    The uniformity and its induced topology are the \textbf{initial uniformity/topology} induced by $\seqi{T}$.
-\end{definition}
-\begin{proof}
-    (1), (U): By \ref{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 \ref{definition:continuous-linear} and (4) of \ref{definition:initial-uniformity}.
-\end{proof}
-
 \begin{definition}[Product Topology]
 \label{definition:tvs-product}
     Let $\seqi{E}$ be TVSs over $K \in \RC$ and $E = \prod_{i \in I}E_i$ be their product as a vector space, and $\fU$ be the initial uniformity generated by the projection maps, then