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An exercise on Fréchet Spaces

Functional Analysis
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Tsakanikas Nickos
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An exercise on Fréchet Spaces


Post by Tsakanikas Nickos » Sat Mar 04, 2017 10:16 pm

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.
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Re: An exercise on Fréchet Spaces


Post by r9m » Mon May 15, 2017 6:35 am

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.
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