Is there a canonical way to make a DF-space Fréchet while keeping the same vectorial structure? Or the converse? I've been looking in the classical books for locally convex spaces but haven't found anything.
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3$\begingroup$ These are essentially disjoint classes in the sense that their intersection consists of the Banach spaces. One can craft a more formal version of this claim but this is the kindergarten version (hint: use the Baire category theorem on the sequence of bounded sets which would exist if it were a $DF$- space). $\endgroup$– klempnerCommented Dec 6, 2022 at 17:51
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1$\begingroup$ klempner's comment is certainly correct in spirit although separable Fréchet and DF-spaces have the same algebraic dimension and can be made algebraically isomorphic. But then the two topological structures are uncomparable because otherwise the closed graph theorem would make them equal. An answer to the question: Do not try to make a Fréchet space DF. $\endgroup$– Jochen WengenrothCommented Dec 9, 2022 at 8:23
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$\begingroup$ The DF space $\phi$ of finitely-supported $\mathbb{R}$-valued sequences has countably-infinite Hamel dimension, but the Hamel dimension of an infinite-dimensional Fréchet space $E$ is uncountable (if $\mathrm{dim}(E)$ were countable we could express $E$ as the countable union of finite-dimensional subspaces, which are closed sets with empty interior, contradicting the Baire category theorem). So $\phi$ is not linearly isomorphic to any Fréchet space, even discontinuously. $\endgroup$– Robert FurberCommented Dec 13, 2022 at 14:58
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