Talk:Abstract Wiener space

Hahn-Banach

Hmm. Should I think of the extension of H to B as an application of the Hahn–Banach theorem? Or is that incorrect? 67.198.37.16 (talk) 18:52, 27 May 2024 (UTC)

Not really, in abstract Wiener spaces you introduce a weaker norm (measurable norm) and then you take completion under this new norm.--Tensorproduct (talk) 12:07, 9 May 2025 (UTC)

Self-contradictory

The article currently makes a pair of contradictory statements. First, in the sub-section titled Cylinder set measure on H it says:

The collection of all cylinder sets forms an algebra of sets in but it is not a σ -algebra.

A paragraph later, in the subsection titled Nonexistence of the measure on H we read this:

...on the σ-algebra generated by the collection of cylinder sets in H.

This is confusing. So which is it? Is it a sigma algebra, or not? Placed in between these two is the following remark:

...although the same set C can be represented as a cylinder set in more than one way,...

What does this mean? How many ways can it be represented? We were just pointed at the Riesz representation theorem, which says the representation is unique. But now it's not unique? Perhaps there is something obvious here, but I don't see it at the moment. 67.198.37.16 (talk) 22:52, 8 May 2025 (UTC)

It's not a contradiction. The cylinder sets form an algebra ${\mathcal {A}}$ but a measure is defined on a σ-algebra, hence if you want to define a measure you have to take the smallest σ-algebra $\sigma ({\mathcal {A}})$ to define a measure, and this is what is meant with the σ-algebra generated by cylinder sets. Your second question: a cylinder set is a set $\{x,\dots \}$ in $H$ such that $(f_{1}(x),\dots ,f_{n}(x))\in E$, this set does not have to have a unique representation in terms of $f_{1},\dots ,f_{n},E$.--Tensorproduct (talk) 11:55, 9 May 2025 (UTC)

I think the issue is that there are multiple distinct concepts, jumbled together with very nearly identical names and wording. The first phrase should be reworded to say:

The collection of all cylinder sets forms an algebra of sets, called the cylindrical algebra. It is not a sigma algebra, but can be used as a basis to generate a sigma algebra.

I would like to link cylindrical algebra to an article that defines it. And, why, there it is! It is defined here: Cylindrical σ-algebra and it's defined in a section you added just a few months ago. Reviewing that, its now clear; the issue was really word choice and sentence construction leading to confusion. I'll try to fix this up now. 67.198.37.16 (talk) 15:37, 9 May 2025 (UTC)

Anyway, I made that change, so this issue is closed, for me. 67.198.37.16 (talk) 16:15, 9 May 2025 (UTC)

A collection of cylinder sets can form a σ-algebra, but in infinite dimension they do not.--Tensorproduct (talk) 17:24, 9 May 2025 (UTC)

Sobolev space

I'm pretty sure that the space $L^{2,1}$ used in the final section, relating classical Weiner space to abstract Wiener space, is isomorphic to the Sobolev space $W^{1,2}$ but I'm not totally sure, because Sobolev spaces are conventionally defined with a slightly different inner product. In the Sobolev case, it would be

$\langle \sigma _{1},\sigma _{2}\rangle _{W_{0}^{1,2}}:=\int _{0}^{T}\langle \sigma _{1}(t),\sigma _{2}(t)\rangle _{\mathbb {R} ^{n}}\,dt+\int _{0}^{T}\langle {\dot {\sigma }}_{1}(t),{\dot {\sigma }}_{2}(t)\rangle _{\mathbb {R} ^{n}}\,dt.$

Note that $W^{1,2}$ is a Hilbert space (see article) and that it has fourier series coefficients that I think obey the same constraints as needed for the conventional fourier series expansion for a fixed Wiener process. So I'm thinking that, for this section, $L^{2,1}$ should be replaced by the more familiar $W^{1,2}$ but I don't know if that's allowed, or if that upsets some apple cart I'm not familiar with. 67.198.37.16 (talk) 04:27, 9 May 2025 (UTC)