Timeline for Nuclear spaces and intuition behind their topology
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 11, 2023 at 12:09 | answer | added | quarta | timeline score: 5 | |
| Mar 7, 2023 at 15:18 | comment | added | Liviu Nicolaescu | Have a look Gelfand and Vilenkin "Generalized functions. Applications of harmonic analysis*. This is the 4th volume of the 5 volume gem about generalized functions by Gelfand and various coauthors. Section I.3 of volume 4 discusses a special, but very general class of nuclear spaces. These examples will cover most concrete needs, and it has the advantage that the arguments are much more transparent than in the general case. | |
| Mar 7, 2023 at 14:02 | comment | added | terceira | The basic intuition is that for a nuclear operator. I think that a good way to think of this is via the analogy in the commutative cases-- bounded operators like $~\ell^\infty$, compact operators $~c_0$, nuclear operators $~\ell^1$ (and on Hilbert space, Hilbert-Schmidt operators $~\ell^2$). This is even exact for sequences acting as diagonal operators on $\ell^2$. | |
| Mar 7, 2023 at 8:34 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Mar 6, 2023 at 23:45 | comment | added | Dmitri Pavlov | Yes, this is exactly what the Wikipedia article is alluding to above: the unit ball of X_q becomes “much smaller” when mapped to X_p, i.e., the unit ball of q is “much smaller” than the unit ball of p. | |
| Mar 6, 2023 at 23:29 | comment | added | user267839 | @DmitriPavlov: If I try to express it in layman's words than the intuition behind $X_q \to X_p$ beeing nuclear seems to be that "typical open sets" like balls with respect to seminorm $q$ look "skinny" trough glasses of the topology induced by seminorm $p$. So in some way the nuclear maps "narrow" the volume of the images of open sets in domain (not in the sense of a "scalling" but more in degrees of freedom sense; I don't know how to express it better; ) making looking them "sparse" comparing with typical open sets of the codomain space. Is this intuition of nuclear maps halfway correct? | |
| Mar 6, 2023 at 22:59 | comment | added | Dmitri Pavlov | This is answered in my comment above. Property 7 is a precise formulation of “much smaller”. Nuclear maps are maps with “small” image, intuitively. | |
| Mar 6, 2023 at 22:56 | comment | added | user267839 | @DmitriPavlov: yes the nuclear maps essentially mimic (in certain sense) the structure of linear map between finite dimensional vspaces $X \to Y$ to be given by the sum $\sum a_i x_i' \otimes y_i$ where for certain $x_i' \in X'$ and $y_i \in Y$ and scalars $a_i$. For sure in infinite dimensions not all maps are of this type. But I not see what does it in this sense mean that the unit ball $B_{1,q}(0) \subset X$ is "musch smaller" than unit ball $B_{1,p}(0) \subset X$. What here the formulation "much smaller" mean? | |
| Mar 6, 2023 at 22:42 | comment | added | Dmitri Pavlov | Your question is already answered by the Wikipedia article that you referenced: Property 7 in the “Characterizations” section make this precise: for any seminorm p p we can find a larger seminorm q q so that the natural map X_q→X_p is nuclear. | |
| Mar 6, 2023 at 22:29 | history | asked | user267839 | CC BY-SA 4.0 |