Talk:Fréchet space

message: I tripped accross the following definition of a Frechet product metric
:d(x,y) = \sum_{n=0}^\infty \frac{1}{2^n} 
\frac{\vert x_n-y_n\vert}{1+\vert x_n-y_n\vert}
as a metric on the product topology of \mathbb{R}^\omega, the product of a countable number of copies of the reals. This metric turns \mathbb{R}^\omega into a Frechet space.  I guess I should add this as an example to the article.  Not sure why I thought to mention it here first. linas 14:46, 21 November 2005 (UTC)
thread_title:  Frechet product metric 

message: From Counterexamples in Topology:
Every separable metric space is homeomorphic to a subspace of Frechet space.  A mapping  is given by the function 
:f(x) = \langle d(x,a_n) \rangle 
where x is a point in the metric space, and the \{a_n\} is a countable dense subset of the metric space. The angle brackets are used to denote a point in Frechet space as y=\langle y_n \rangle.  
I suppose I should add the above to the article, but I thought I'd mention it here first. linas 15:23, 21 November 2005 (UTC)
replies: text: I don't understand what the angle brackets mean, nor can I understand your explanation. -lethe talk 15:55, 21 November 2005 (UTC)
replies: text: I looked in Counterexamples, and they apparently use the angle brackets to denote ordered families, as in products.  So OK. -lethe talk 16:40, 21 November 2005 (UTC)
thread_title:  Another example 

message: Does C∞(R) become a Fréchet space if we use the countable set of seminorms 
:\|f\|_{m,n} = \sup\{ |f^{(n)}(x)| : x\in [-m,m]\}
for m,n ∈ N? AxelBoldt (talk) 05:54, 2 February 2009 (UTC)
replies: text: Yes, of course.  If a sequence of C^\infty functions on R is Cauchy for this family of semi-norms, then for every given segment, the nth and n+1 derivatives converge uniformly on that segment, hence you prove, by induction on n, that the limit of the n+1 derivatives is the derivative of the limit of n-th derivatives, by the classical theorem for the C^1 case. --Bdmy (talk) 07:46, 2 February 2009 (UTC)
replies: text: Thanks! AxelBoldt (talk) 02:28, 3 February 2009 (UTC)
thread_title: C<sup>&infin;</sup>('''R''') ?

message: The equivalence of the two definitions for Fréchet spaces given in this article requires that all topological vector spaces are implicitly assumed to be Hausdorff. The definition in topological vector space is however ambiguous on that question. So I think we either have to clarify the definition in that article, or add the Hausdorff requirement to this one. AxelBoldt (talk) 02:26, 3 February 2009 (UTC)
thread_title:  Topological vector spaces are Hausdorff? 

message: Regarding [this edit]: I don't think "complete" means anything for general topological spaces. The redirect Complete space also mentions this in its edit history. YohanN7 (talk) 12:22, 4 April 2017 (UTC)
replies: text: Note that there is no natural notion of distance between two points of a Fréchet space: many different translation-invariant metrics may induce the same topology.
Of course. The same comment can be made about any metrizable space. I think we have two inequivalent definitions of Fréchet spaces at hand. One has a metric, the other has not. YohanN7 (talk) 12:29, 4 April 2017 (UTC)

text: Fair enough. (You mean the first paragraph in Complete metric space#Alternatives and generalizations, right?) YohanN7 (talk) 14:30, 5 April 2017 (UTC)

text: Right.  ―Toby Bartels (talk) 02:59, 9 April 2017 (UTC)

text: \bigcap_{p \in \mathcal P} \{x \in X: p(x) = 0\} = \{0\}.
More unrelated: Lead; the Hahn-Banach theorem holds, as far as I know, in general vector spaces. no further structure needed. YohanN7 (talk) 07:12, 6 April 2017 (UTC)

text: At any rate, some clarification is needed. The Hahn-Banach theorem per se doesn't even need for its proof a topology. The useful corollaries, however, need some more flesh. It should be put that way. I'll have a look at what my reference (Conway) says. He has very little about Fréchet spaces, but says lots about locally convex spaces in general. YohanN7 (talk) 08:20, 19 April 2017 (UTC)
thread_title:  Do Fréchet spaces (constructed using a metric) have a metric or not?