Talk:Nuclear space

message: 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)
thread_title: Definitions

message: The article states: If p is a seminorm on V, we write Vp 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)
replies: text: 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)
thread_title:  Banach spaces with seminorms? 

message: "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)
thread_title:  Definition 

message: 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)
thread_title:  Good example? 

message: Some of the notation in the Motivation section is left unexplained.
Please do not force readers to click on links to find out how the most recent editor happens to have decided to use notation.
For instance, what is \mathcal{D}^{\prime}? We are not told.  
Grothendieck's mathematics is hard enough without having to decipher unexplained symbols. 2601:200:C000:1A0:84C1:C760:8651:1BB5 (talk) 23:02, 16 May 2022 (UTC)
thread_title:  Please state what the notation means!