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At Fréchet space I have added to the Idea-section a paragraph motivating the definition via families of seminorms from the example of ℝ∞=lim⟵nℝn. And I touched the description of this example in the main text, now here.
(Beware that the same symbol “ℝ∞” is also used for the corresponding direct sum/injective limit, which is different.)
Also it is used for the same (“projective”) limit but with ℝn being a dicrete space (what leads to a linearly compact vector space). I added redirect linearly compact vector space to linearly compact module.
I have added the statement (here) that “path smooth” linear functions on a Fréchet space are continuous.
(This sounds like the kind of statement somebody here would already have recorded years back, but I didn’t find it stated in any entry.)
Added:
There are two inequivalent definitions of Fréchet spaces found in the literature. The original definition due to Stefan Banach defines Fréchet spaces as metrizable complete topological vector spaces.
Later Bourbaki (Topological vector spaces, Section II.4.1) added the condition of local convexity. However, many authors continue to use the original definition due to Banach.
The term “F-space” can refer to either of these definitions, although in the modern literature it is more commonly used to refer to the non-locally convex notion.
@Dmitri
what’s a recent reference that uses the older definition?
Re #6:
Albert Wilansky, Modern Methods in Topological Vector Spaces (1978; reprinted by Dover in 2013)
Lucien Waelbroeck, Topological Vector Spaces and Algebras (1971)
(For the subject of topological vector spaces, 1970s could qualify as “recent”.)
Added references for the terminology:
There are two inequivalent definitions of Fréchet spaces found in the literature. The original definition due to Stefan Banach defines Fréchet spaces as metrizable complete topological vector spaces. The books of Waelbroeck (1971) and Wilansky (1978) use the original definition.
In 1953 Bourbaki (Topological vector spaces, Section II.4.1) added the condition of local convexity. This convention is followed by the books of Jarchow (1981), Köthe (1969), Schaefer (1971).
The term “F-space” can refer to either of these definitions, although in the modern literature it is more commonly used to refer to the non-locally convex notion. The books of Köthe (1969) and Schaefer (1971) require F-spaces to be locally convex.
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