Talk:Weak convergence (Hilbert space)

Definition?

An article on wiki covering a topic of mathematics should always contain a formal definition. If it does not, it is s.it, not an article. It is as if an article on Newton's laws of motion contained only the first sentence "Newton's laws of motion are three physical laws that, together, laid the foundation for classical mechanics." No formulation of the laws and then continued with their properties. Wouldn't it be proposterous? Yes it would. Would it be a good article? No, it would be a s.it article. — Preceding unsigned comment added by 92.52.23.13 (talk) 11:09, 9 October 2015 (UTC)

Weak convergence of orthonormal sequences

The current proof assumes that the sequence converges weakly (and then shows that it has to converge to 0 in this case). But how do you know that the sequence converges weakly in the first place?

Also, is the weak limit unique in general (in some suitable sense of what unique means)? Simon Lacoste-Julien 22:18, 25 March 2006 (UTC)

the version i am looking at does not do what you say. the claim is that any orthonomal sequence converges weakly and its weak limit is the 0 vector. and yes, the weak limit is unique, as the weak topology is Hausdorff. Mct mht 13:33, 28 June 2006 (UTC)

Weak continuity

This might be a good place to define weak continuity as continuity on the weak topology. This is a difficult definition to find online. Gheckel (talk) 05:55, 10 March 2008 (UTC)

Example?

I'm still new to this topic so I don't feel confident adding it myself, but I think this example might help the article: fn = sin(nx) converges weakly to f=0 on [0,1]. There is a proof (not mine) here:

http://mathoverflow.net/questions/10186/what-are-some-interesting-sequences-of-functions-for-thinking-about-types-of-conv/10214#10214

This example is good since we could make a nice visual for the article that would show just how weak this type of convergence can be!

-futurebird (talk) 02:40, 31 December 2009 (UTC)

Definitely, fn = sin(nx) converges weakly to f=0 on [0,1] (or rather, in $L_{2}[0,1]$); do not hesitate including this example. If in doubt about some detail, ask me here. Boris Tsirelson (talk) 19:44, 31 December 2009 (UTC)

More generally, if $f_{n}(x)=\sin(\lambda _{n}x)$ for some $\lambda _{n}\to \infty$ (not necessarily integers) then $f_{n}$ converge weakly to 0 in $L_{2}[a,b]$ whenever $-\infty <a<b<\infty .$ Boris Tsirelson (talk) 19:50, 31 December 2009 (UTC)

Still more generally, $\sin$ may be replaced by any continuous (or just locally integrable) periodic function whose integral over the period vanishes. Boris Tsirelson (talk) 05:58, 1 January 2010 (UTC)

I made an image. I just used sin nx since everyone knows it.

The first 3 functions in the sequence $f_{n}(x)=\sin(nx)$ on $[0,2\pi ]$. As $n\rightarrow \infty$ $f_{n}$ converges weakly to $f=0$.

Some of that latex.. looks goofy... futurebird (talk) 06:48, 1 January 2010 (UTC)

Nice. Just do it. Boris Tsirelson (talk) 09:51, 1 January 2010 (UTC)

In fact, your case is a special case of "Weak convergence of orthonormal sequences", since these sin functions are orthogonal. They are not normalized, but this is a matter of a constant coefficient, which does not harm. Boris Tsirelson (talk) 10:56, 1 January 2010 (UTC)