Talk:Characterization of Continuity of Linear Functional in Weak-* Topology

Characterization of Continuous Linear Functionals in Weak-* Topology? Caliburn (talk) 21:19, 16 June 2023 (UTC)

Singular noun -- the theorem applies to an arbitrary continuous linear functional. "Characterization of" suits well enough, despite our aim to avoid words that are different either side of the pond. But I thin k I'm okay to be firm on the singularness of the noun.

Same applies to the other page. It's a style, it's near enough 100% consistent throughout, nice to sustain that vibe. --prime mover (talk) 23:07, 16 June 2023 (UTC)

And yep, I seem to have over-generalised without thinking enough. $X^\ast$ has not been given a topology here. We can of course define the $w^\ast$ topology still, but I think there is a more standard strong topology called the polar topology. I don't know the details so I will just scale back this page to an NVS. Caliburn (talk) 22:08, 16 June 2023 (UTC)

Do you mean the weak topology? So, is the equality $\struct {X^\ast, w^\ast}^\ast = \struct {X^\ast, w}^\ast$ true? --Usagiop (talk) 22:43, 16 June 2023 (UTC)

No, explicitly:

$\struct {X^\ast, w^\ast}^\ast = \set {x^\wedge : x \in X} = \iota X$

as sets, (there is no topology given to the LHS) where $\map {x^\wedge} f = \map f x$ for each $f \in X^\ast$. As to $\iota X$ vs $\iota \sqbrk X$, I'm not sure. I guess it's whatever looks better. I also would not object to going back to writing $J$ or $j$ instead, as is often found in literature.

The equality you write is true if and only if $X$ is reflexive, since it is precisely the statement that $\iota X = X^{\ast \ast}$. (the weak dual is the same as the norm dual) This is essentially proven in Normed Vector Space is Reflexive iff Weak and Weak-* Topologies on Normed Dual coincide. Caliburn (talk) 22:48, 16 June 2023 (UTC)

OK, so you mean:

$\struct {X^\ast, w^p}^\ast = \iota X$

where $w^p$ is the polar topology, should be true. --Usagiop (talk) 23:11, 16 June 2023 (UTC)

I don't think I do? I definitely mean the $w^\ast$ topology on $X^\ast$. Then the only continuous linear functionals $\struct {X^\ast, w^\ast} \to \GF$ are the evaluation maps. I don't know anything about the polar topology yet. This is Theorem 3.16 in Functional Analysis and Infinite-Dimensional Geometry and it appears precisely like this. Caliburn (talk) 23:15, 16 June 2023 (UTC)

OK, I just misunderstood. You then only wanted to mention the polar topology of $X^\ast$, nothing more. --Usagiop (talk) 23:24, 16 June 2023 (UTC)

I thought the polar topology was something different. Caliburn (talk) 23:25, 16 June 2023 (UTC)

So you mean you think:

$\struct {X^\ast, w^p}^\ast \ne \iota X$

--Usagiop (talk) 23:27, 16 June 2023 (UTC)

To be clear, my problem was that I wrote $X^{\ast \ast}$ (meaning continuous linear functionals on $X^\ast$) without having given $X^\ast$ a topology. Caliburn (talk) 22:56, 16 June 2023 (UTC)

Also it says if and only if so we get the equality. Caliburn (talk) 22:13, 16 June 2023 (UTC)

I'm a little worried about unsourced stuff here. From the discussions going back and forth between Caliburn and Usagiop (here and elsewhere) makes me wonder whether this stuff is being generated on the fly, so to speak. I would counsel against this. We have had other instances where a contributor has forged ahead off his own bat without being completely certain what's being talked about.

Sources are essential at this level, because they have already covered the concepts. OTOH unless those sources are studied in depth and their exposition traced through alongside the existing pages, it is never a sure thing that what they say is actually what the pages on $\mathsf{Pr} \infty \mathsf{fWiki}$ say -- all of which need to be reported precisely.

My own education ceased at elementary topology and all this functional analysis is more or less opaque to me, because the nature of the objects involved is obscured by the technical language surrounding it, none of which is accessible or even consistent. Hence my plea to keep it accessible, and rooted in the solid earth of definitions traceable back to ZFC and PredLog.

Hence it may be worthwhile trying to add something to these pages somewhere which acts as either a motivator or a depictionator of some kind. For example, we have a dual, and a double dual, both defined in a number of different contexts, but with nothing bringing it conceptually together to explain this in intuitively accessible terms. --prime mover (talk) 00:00, 17 June 2023 (UTC)

I assume this page is a lemma, which will be used in several proofs. So, this result is actually opaque. By the way, it may be worth to create a page to list up all dual space concepts. --Usagiop (talk) 00:52, 17 June 2023 (UTC)

Having slept on it, $\struct {X^\ast, w^\ast}^\ast = \set {x^\wedge : x \in X}$ is strictly speaking still correct for $X$ an arbitrary topological vector space, (and is briefly stated in Rudin) this result really is just a trivial and uninteresting consequence from a general theorem on initial topologies on vector spaces and does not need any structure at all on the vector space beyond the maps separating points. (so the generated topology is actually that of a TVS) I need this result for two reasons: to prove that a map $f : \struct {X, \tau} \to \struct {Y^\ast, w^\ast}$, I apply this result with Continuity in Initial Topology to argue that all I need to do is show that $y^\wedge \circ f : \struct {X, \tau} \to \GF$ is continuous for each $y \in Y$. Also, if I know a linear functional $\Phi : \struct {X^\ast, w^\ast} \to \GF$ is continuous, I quickly get $\Phi = x^\wedge$.

The problems: I only define the map $\iota$ for normed vector space, and reflexivity is typically discussed within the context of NVSs. I also wrote $X^{\ast \ast}$ which doesn't make sense as I intended it without a topology. I won't generalise to TVSs without knowing exactly what I'm doing, it was just an instance of me going too quickly and generalising the result $\struct {X, w}^\ast = X^\ast$, which does hold in arbitrary TVSs without any definitional complications. I have checked other links to "Topological Dual Space" and I haven't made this error elsewhere.

As to the polar topology stuff, I don't know what's going on. I don't need any of it and it hasn't come up in any of the sources I am working through. I just mentioned it because it's a topology that can be placed on $X^\ast$ when $X$ is not a normed vector space but might have some other nice structure. For the Banach-Alaoglu stuff, I made an offhand remark about Eberlein–Šmulian having just became aware of it. Then we had a disagreement over what the correct statement of it is, though I then found in a book that one statement implies the other in a non-trivial way hence the confusion.

Most of this is being done from my functional analysis lecture notes and problem sheets only consulting books for definitions, but there is a book "Functional Analysis and Infinite Geometry" that covers much of the same content and I have put references in in places. It's all very conceptually difficult juggling different topologies. It got even worse in my course, because $X$ was identified as a subset of $X^{\ast \ast}$, and you had to mentally juggle inbetween considering $x \in X$ and $\iota x \in X^{\ast \ast}$ within the same proof. I won't do any such identification here and will always make explicit $x^\wedge$ (when we're thinking of it as an evaluation map) or $\iota x$ (when we want to think of it as the image of $x$). I am welcome to suggestions on notation, though. I changed from $J$ to $\iota$ and I already sort of regret it. Caliburn (talk) 09:04, 17 June 2023 (UTC)

The notation like $\struct {X^\ast, w^\ast}^\ast$ looks like suggesting a topology, but in fact, no topology is specified. $\struct {X^\ast, w^\ast}$ is a topological space, but $\struct {X^\ast, w^\ast}^\ast$ is just a set. The $w^\ast$ is only used to specify the outer $^\ast$. The conclusion here is also an equality as sets, not as topological spaces. I think, the notation is really confusing, but I have no good ideas. --Usagiop (talk) 10:19, 17 June 2023 (UTC)

Yep, well to be exact equality as plain vector spaces. I have no alternative to suggest. Rudin doesn't seem to give $X^\ast$ a topology beyond the $w^\ast$ one. Wikipedia says we could put the strong dual topology on it, but I will wait until I encounter it in a book. Caliburn (talk) 10:31, 17 June 2023 (UTC)

Also I will delete the explain - the evaluation maps are continuous by the definition of the $w^\ast$ topology, so the other inclusion is trivial. The question is whether making the evaluation maps continuous creates any more continuous linear functionals, and this theorem answers that question in the negative. Regardless, Continuity of Linear Functionals in Initial Topology on Vector Space Generated by Linear Functionals is an if and only if, and builds this in. I will add a sentence "recalling that the $w^\ast$ topology is generated by the evaluation maps". Caliburn (talk) 09:09, 17 June 2023 (UTC)

IMHO, you should either only state $\subseteq$, or give a proof for both $\subseteq$ and $\supseteq$. It is not such trivial, because of the visual ambiguity to understand what you mean with the different $^\ast$s, $\iota$ and $^\wedge$. --Usagiop (talk) 10:19, 17 June 2023 (UTC)

if and only if is given in Continuity of Linear Functionals in Initial Topology on Vector Space Generated by Linear Functionals, I don't understand what the problem is. Caliburn (talk) 10:29, 17 June 2023 (UTC)

Sorry, just my mistake. --Usagiop (talk) 13:35, 17 June 2023 (UTC)

No worries, this area in general is inherently a notational mess. Caliburn (talk) 13:37, 17 June 2023 (UTC)

And of course I have to state that $\phi$ is a linear functional. Sorry, this was like a one sentence page I put up in the middle of writing another proof. I will review my other pages. Caliburn (talk) 09:12, 17 June 2023 (UTC)