An exercise on Fréchet Spaces
-
- Community Team
- Posts: 314
- Joined: Tue Nov 10, 2015 8:25 pm
An exercise on Fréchet Spaces
Let $V,W$ be Fréchet spaces and let $T$ be a Hausdorff space. Consider the diagram
\[ V \overset{f}{\longrightarrow} W \overset{i}{\longrightarrow} T \]
where $i$ is a continuous, linear, injective map and $f$ is a linear map. Show that $f$ is continuous if and only if $ i \circ f $ is continuous.
\[ V \overset{f}{\longrightarrow} W \overset{i}{\longrightarrow} T \]
where $i$ is a continuous, linear, injective map and $f$ is a linear map. Show that $f$ is continuous if and only if $ i \circ f $ is continuous.
Re: An exercise on Fréchet Spaces
Since, $T$ is Hausdorff, the graph of the continous linear map $i \circ f$ is closed subspace of $V \times T$. Again, the linear map, ${Id}_{V} \times i : V \times W \to V \times T$ is continuous and $({Id}_{V} \times i )^{-1} (G_{i \circ f}) = G_f$ (since, $i$ is a injective map).
\begin{align*}G_{f} \subseteq V \times W \overset{{Id}_{V} \times i}{\longrightarrow} G_{i \circ f} \underset{\text{(closed)}}{\subseteq} V \times T\end{align*}
Hence, $G_f$ is a closed subspace of $V \times W$, i.e., by Closed Graph theorem it follows that $f$ is continuous.
\begin{align*}G_{f} \subseteq V \times W \overset{{Id}_{V} \times i}{\longrightarrow} G_{i \circ f} \underset{\text{(closed)}}{\subseteq} V \times T\end{align*}
Hence, $G_f$ is a closed subspace of $V \times W$, i.e., by Closed Graph theorem it follows that $f$ is continuous.
Create an account or sign in to join the discussion
You need to be a member in order to post a reply
Create an account
Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute
Sign in
Who is online
Users browsing this forum: No registered users and 1 guest