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 $X$ 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 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 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)