Talk:Locally convex topological vector space

message: Just started with the basics here (it was on Wikipedia:Requested articles/mathematics).  Most of the Topological vector space article is actually on LCTVSs, so I think rather than take this page much further on its own, some material be transferred.  --AndrewKepert 03:30, 11 Mar 2004 (UTC)
replies: text: PS.  I had been considering moving chunks of stuff from the TVS article to here, but on second thoughts, I don't think it (the TVS article) should be gutted.  It is the most general class here, so to have an overview of everything that happens within it (all the way down to the most special, such as Hilbert spaces) is sensible.  It could do with a reorganise though.  -- Andrew Kepert 07:34, 4 May 2004 (UTC)

text: OK thanks for picking that up - you are right.  When I wrote the page I didn't check that all links were appropriate.  I will fix.  -- Andrew Kepert 04:33, 11 May 2004 (UTC)

text: OK, created Seminorm -- Andrew Kepert 06:27, 11 May 2004 (UTC)
thread_title: Started

message: The second phrase of the "Definition" is a bit confusing, especially the "this" seems to refer to an already given local basis, while there is clearly no connection between the newly introduced family N and the existing topology of the TVS. I will try to clarify; please check if you agree to changes. — MFH: Talk 18:32, 21 Apr 2005 (UTC)
replies: text: I clarified the construction. Is it clearer now ? MathMartin 14:48, 24 Apr 2005 (UTC)
thread_title:  local basis from seminorms not clear 

message: I object somehow against changing to the "more specific category" of function spaces;
even if this is the main application, nothing requires /a priori/ a LCTVS to be
a function space (in the usual sense --- of course, one could say that even the real
numbers are also a function space, when seen as quotient of sequence spaces...) — MFH: Talk 22:45, 19 September 2005 (UTC)
replies: text: Yes, I guess I was wondering if this objection might be raised. I can put it back, but need your help with one thing, then. The :Category:Functional analysis has 120 articles in it, and I was hoping to create a subcategory into which some of these could be moved. I was thinking along the lines of "those articles having more of a topological slant than usual" as being the subcategory, but admit this is rather vague and poorly defined. Any suggestions for a better split? linas 23:58, 19 September 2005 (UTC)
replies: text: OK I'll put it back.  linas 03:17, 23 September 2005 (UTC)
thread_title:  too specific category 

message: Is the product of two LCTVS's also a LCTVS (assuming the product topology) ?  Seems to be so for Frechet spaces, not clear if this is a general property. linas 03:17, 23 September 2005 (UTC)
replies: text: Sure it is.  Just take take the union of the two families of seminorms with  pα(x,y)=pα(x) and qβ(x,y)=qβ(y).  A seminorm is, after all just projection onto one component, and a product is just a space with extra components.  I expect that only countable products of Fréchet spaces will be Fréchet spaces.  An uncountable product might not have a countable family.  I'm not sure I can prove it doesn't though, so maybe you're right, and any product of Frechet is Frechet. -lethe talk 04:06, 8 January 2006 (UTC)
thread_title: Product spaces

message: Elroch, you write "Revert. I think my edit was misread - the circle is specified as being the one *in* the 1-dimensional vector space generated by x".  Right you are.  Apologies. -lethe talk [ +] 02:28, 4 May 2006 (UTC)
thread_title:  misread edit 

message: I prefer the definition of "balanced" that was previously in the article, as it is the most obvious use of the word, but a check of various sources indicates there is agreement on the modified definition I have added. The previous definition is now called "circled". Elroch 14:06, 4 May 2006 (UTC)
thread_title: Terminology

message: Two comments from someone who doesn't know enough to edit this article but read it hoping to learn from it:
1. The definition of "absorbent set" given in the article contradicts the definition given in the article on absorbent sets.
2. I was hoping to find some good examples of non-locally convex spaces.  Unfortunately, the article didn't provide enough details about the two non-examples for me to see understand why they are non-examples.  I know case of the Lp spaces for p<1 is discussed in some detail in Rudin's _Functional Analysis_, but I have not seen the second example elsewhere, and it looks non-trivial to work out the details based on the remarks in the article.  —Preceding unsigned comment added by 128.237.251.63 (talk) 04:24, 2 October 2007 (UTC)
thread_title:  absorbent sets, examples of non-locally convex spaces 

message: Given the terminology, it seems very strange not to give the "obvious" definition: a TVS with a base of convex sets.  If desired, one can remark that the balanced and absorbent conditions can also be required (but are redundant).  Am I missing something?
128.178.14.170 (talk) 08:45, 2 March 2010 (UTC)
replies: text: I think so. Why do you think these properties are redundant?  The "absorbent" property is needed to exclude e.g. the discrete topology on a C-vector space. As for "balanced", perhaps you made the same mistake I initially made, thinking that any convex set including the origin is automaticlaly balanced? This is not so, take e.g. the interval [-1,2] in R, which is not balanced. And the topology given on R by making the sets [0, 1/n] a basis of neighbourhoods of 0 is convex, but not balanced. (Neither is it absorbent.) I'm still thinking on an example of a convex absorbent, non-balanced space... --Roentgenium111 (talk) 00:18, 9 April 2010 (UTC)
replies: text: Quoting Rudin, every open neigborhood of 0 contains a balanced neighborhood of 0 and every convex neighborhood of 0 contains a balanced convex neighborhood of 0 [Thm.1.14]. Also, every open neigborhood of 0 is absorbent [Thm.1.15a], so non-balanced or non-absorbent TVS or LCTVS make no sense. Lapasotka (talk) 18:53, 29 July 2011 (UTC)
thread_title:  Change the definition? 

message: Beside the general definition (see previous comment), I also find the following a bit misleading:
"Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn-Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals."
Why not simply give it straight: local convexity is *equivalent* to the natural Hahn-Banach theorem. Normability is a distraction.  —Preceding unsigned comment added by 128.178.14.170 (talk) 08:49, 2 March 2010 (UTC)
thread_title:  Another improvement that should be made 

message: I think there is something wrong with the following passages. The parts I have problem with are underlined.
A Cauchy net  in a locally convex space is a net {xκ}κ such that for every ε > 0 and every seminorm pα, there exists a κ such that for every λ, μ > κ, pα(xλ–xμ) < ε.  In other words, the net must be Cauchy in all the seminorms simultaneously.  The definition of completeness is given here in terms of nets instead of the more familiar sequences because unlike Fréchet spaces which are metrisable, general spaces may be defined by an uncountable family of pseudometrics.    A locally convex space is complete if and only if every Cauchy net converges.
* An important function space in functional analysis is the space D(U)  of smooth functions with compact support in U ⊆ Rn.   A more detailed construction is needed for the topology of this space because the space C∞0(U) is not complete in the uniform norm. The topology on D(U) is defined as follows: for any fixed compact set K ⊂ U, the space C∞0(K) of functions ƒ ∈ C∞0(U) with supp(ƒ) ⊂ K is a Fréchet space with countable family of seminorms ||ƒ||m = supx |Dmƒ(x)| (these are actually bona fide norms, and the space C∞0(K) with the ||·||m norm is a Banach space Dm(K)) .  Given any collection {Kλ}λ of compact sets, directed by inclusion and such that ∪λKλ = U, then the C∞0(Kλ) form a direct system, and D(U) is defined to be the limit of this system.  Such a limit of Fréchet spaces is known as an LF space.  More concretely, D(U) is the union of all the C∞0(Kλ) with the final topology which makes each inclusion map C∞0(Kλ)↪D(U) continuous.    The space is not metrisable, and so it is not a Fréchet space.  The dual space of D(Rn) is the space of distributions on Rn.
As far as I understand, D(U) is an example of a non-metrizable complete locally convex vector space. (Non-metrizability implies that the topology D(U) is not induced by a countable family of seminorms.) Furthermore, it seems that D(K) is a Frechét-Urysohn space, which means that closures of its subsets coincide with their sequential closures.
Proof. Let f be in the closure of A⊂D(U). Then f∈C∞0(K) for some compact set K⊂U. Since C∞0(K)⊂D(U) is closed, the C∞0(K)-closure of A∩C∞0(K) coincides with its D(U)-closure. Since C∞0(K) is metrizable, there exists {fj}j=0∞⊂C∞0(K)∩A such that fj→f in C∞0(K), and hence also in D(U). This proves that D(U) is Frechét-Urysohn.□
This indicates that the topology of D(U) is quite well characterized by convergent sequences. Note that we only used the fact that a sequence {fj}j=0∞ converges to f in D(U) if and only if there exists a compact K⊂U which contains the support of every fj and fj→f in C∞0(K). So how about the Cauchy-nets?
Assume that every Cauchy-sequence in a topological vector space X converges. If there exists a Cauchy-net {xκ} of X which does not converge, then there exists a neighborhood U⊂X of 0 such that for every κ there exists λ>κ with fλ∈X\U. Hence we can pick a Cauchy-sequence {xκ(j)}j=0∞⊂{xκ} which does not converge in X, a contradiction.
Am I missing something? If not, these passages should certainly be modified. Lapasotka (talk) 15:30, 27 July 2011 (UTC)
thread_title:  On completeness 

message: I got redirected from locally convex basis to here but we do not have a single occurence of the word basis in the whole article...
Should there be a new article about convex basis? At least we should kill the redirect, shouldn't we? IXhdBAH (talk) 12:37, 12 December 2011 (UTC)
thread_title:  Redirect from [[Locally convex basis]] 

message: "this means that all Cauchy sequences converge, but not necessarily more general Cauchy nets"
But Treves, Topological Vector Spaces, Distributions and Kernels (Theorem 13.1) says that all (strict) LF spaces are complete, and in Example II that follows, it says that the space of test functions is LF space. 18.111.33.87 (talk) 04:51, 22 January 2013 (UTC)
thread_title:  Is the space of test functions complete? 

message: The article first gives one definition. Then it gives another. Then it establishes a partial relationship. Then it sort of finishes establishing the relationship.
But the phrasing is all wrong. The first definition is introduced as definitive. The second definition is introduced as if definitive. Then an implication is proven between them, without apology for previously treating the definitions as definitive. Then the bidirectional relationship is introduced for the first time.
The article does not have the clarity of any normal normal math text. Instead, it reads like a babble of different voices interrupting one another -- a collection of undergraduates sorting out the definitions! 
Today's winner of Wikipedia's Caveat Lector award. 84.227.20.67 (talk) 09:58, 21 August 2015 (UTC)
replies: text: Okay, it wasn't  as bad as I said. And on top of that, I improved it (I hope). 84.227.20.67 (talk) 10:38, 21 August 2015 (UTC)
thread_title:  Confused definition of "Locally convex topological vector space" 

message: The current definition assumes that the field is ordered without saying so, while some care is taken in the other properties to make clear that this isn't always the case. Shouldn't / can't the " > 0 " be omitted -- it does not change much for ordered fields [actually, the current definition of absorbent set implies this and more ! (see also earlier critics about "absorbing", above)], and the definition would becomes applicable and useful for non-ordered fields. I will tentatively make an edit in that sense. - — MFH:Talk 15:01, 27 June 2018 (UTC)
thread_title:  absorbent set 

message: The first paragraph in the History section reads as follows:
"Metrizable topologies on vector spaces have been studied since their introduction in Maurice Fréchet's 1902 PhD thesis Sur quelques points du calcul fonctionnel (wherein the notion of a metric was first introduced). After the notion of a general topological space was defined by Felix Hausdorff in 1914 although locally convex topologies were implicitly used by some mathematicians, up to 1934 only John von Neumann would seem to have explicitly defined the weak topology on Hilbert spaces and strong operator topology on operators on Hilbert spaces. Finally, in 1935 von Neumann introduced the general definition of a locally convex space (called a convex space by him)."
Can someone please explain the point of this claim:
"[U]p to 1934 only John von Neumann would seem to have explicitly defined the weak topology on Hilbert spaces and strong operator topology on operators on Hilbert spaces."
Only one person defined those two mathematical concepts? How many people should it have been?
And von Neumann only "would seem to have" explicitly defined those things?  Or is it established fact?
I do not understand what the point is.108.245.209.39 (talk) 22:46, 5 August 2018 (UTC)
thread_title:  What is the point?