Timeline for Fréchet and DF spaces

6 events

when toggle format	what		by	license	comment
Dec 13, 2022 at 14:58	comment	added	Robert Furber		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.
Dec 9, 2022 at 8:23	comment	added	Jochen Wengenroth		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.
Dec 6, 2022 at 17:51	comment	added	klempner		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).
Dec 6, 2022 at 15:56	history	edited	Daniele Tampieri	CC BY-SA 4.0	Minor corrections
Dec 6, 2022 at 15:39	history	edited	YCor		edited tags
Dec 6, 2022 at 15:11	history	asked	Mar	CC BY-SA 4.0