Talk:Nuclear space

Definitions

This is a poorly formulated article, and I am not enough of an expert on the topic at this time to improve it. The article states, "Although important, nuclear spaces are not widely used, possibly because the definition is notoriously difficult to understand," but does little to relax this difficulty. For example, Definition 1 states that a space is nuclear if certain maps are nuclear, and states, "The condition of being a nuclear operator is subtle, and more details are available in that article." Instead of being illuminating, this is an encyclopedic mess. The entry on nuclear operators begins with a discussion of nuclear Hilbert space operators (i.e., trace-class operators), which is not what this article should be about, namely, nuclear operators on nuclear spaces.

Improving this article would be a nice exercise for a graduate student in functional analysis. Here is a checklist of some things to be improved:

Add 1-3 paragraphs to the preamble of the section, stating intuitively what is a nuclear space, as well as summarizing the general concept of a nuclear operator.

There should be a section with the technical definition of a nuclear operator, since the definition of a nuclear space references this. This definition does not need to be comprehensive (that's what the nuclear operator article is for), but it should at least be precise.

Be consistent with definitions. If the article is to state six distinct definitions of a nuclear space, then there should be a theorem proving why these definitions are equivalent. As stated, these definitions are not equivalent. Definition 2 deals with Hilbert seminorms, whereas nuclear spaces are defined simply be seminorms which need not be Hilbert.

Separate the content on nuclear Hilbert spaces from general nuclear spaces. Perhaps present the Hilbert material first to help the reader develop intuition, then present the general theory as developed by Grothendieck.

--Hierarchivist (talk) 14:14, 11 March 2013 (UTC)[reply]

Hilbert space

The article states:

There are no Banach spaces that are nuclear, except for the finite dimensional ones.

I'm naive, but isn't every Hilbert space going to be nuclear? Err, well, at least trace class? Or am I confused? And since Hilbert spaces are Banach spaces, the above statement sounds false to me. I'm confused ... linas 23:07, 24 October 2005 (UTC)[reply]

No, infinite dimensional Hilbert spaces are not nuclear. The identity operator of such a space is not trace class (or even compact). R.e.b. 01:09, 25 October 2005 (UTC)[reply]

Err, OK, thanks. I guess I was confused; I'll have to study a bit. linas 15:00, 25 October 2005 (UTC)[reply]

Another confusing point that I ran into (not an expert in the field, btw):

"If p is a seminorm on V, we write Vp for the Banach space given by completing V using the seminorm p."

Hence I conclude that the completion of V with respect to a norm or seminorm is a Banach space.

"There are no Banach spaces that are nuclear."

So the completion of V is not nuclear... so far so good.

"The completion of a nuclear space is nuclear (and in fact a space is nuclear if and only if its completion is nuclear)."

... right. Can anyone help me out here? Are there two meanings of completion? Moocowpong1 00:31, 17 September 2007 (UTC)[reply]

I may have an answer to my own question... is the completion in the third quote referring to this meaning, while the first two are the completion with respect to a seminorm, not the topology of the space? Moocowpong1 06:49, 27 September 2007 (UTC)[reply]

Banach spaces with seminorms?

The article states: If p is a seminorm on V, we write V_p for the Banach space given by completing V using the norm p. Now hold up, p is a seminorm, not a norm, so how on Earth is it going to make a Banach space? -lethe ^{talk +} 19:17, 7 March 2006 (UTC)[reply]

The process of completing a space automatically kills off the norm 0 vectors, so a seminorm on a space is (or more precisely induces) a norm on the competion. (Anyone confused by this is allowed to kill the norm 0 vectors before taking a completion.) R.e.b. 19:54, 2 April 2006 (UTC)[reply]

Link from "nuclear" ?

It would be useful to have a link from the disambiguation page of "nuclear" to this page (and to the page on nuclear operators). Sorry, but I myself don't know how to do that...

Definition

"for any seminorm p we can find a larger seminorm q" What is the definition of "larger" seminorm? --Anton (talk) 22:08, 1 February 2011 (UTC)[reply]

Good example?

Is really $\sin(x+y):\mathbb {R} ^{2}\to \mathbb {R}$ an example of strict inclusion of sets $C^{\infty }(\mathbb {R} )\otimes C^{\infty }(\mathbb {R} )\subset C^{\infty }(\mathbb {R} ^{2})$ ? Isn't $C^{\infty }(\mathbb {R} )\otimes C^{\infty }(\mathbb {R} )$ the linear span of elements of the form $f\otimes g$ ? If so, then $\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)\in C^{\infty }(\mathbb {R} )\otimes C^{\infty }(\mathbb {R} )$ . --Md2perpe (talk) 17:27, 30 April 2021 (UTC)[reply]