Talk:Function of several complex variables: Difference between revisions

Revision as of 13:41, 14 December 2020

Revision as of 13:40, 15 December 2020

Line 74:

:::::Looking at the [[Compact space#example|example of Compact space]], it seems that there is an example of a bounded closed set i.e. unit ball that does not become compact in infinite dimensions. I think I missed the condition of compact set. I also likely need to read the references on the page where the example is shown.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 12:20, 12 December 2020 (UTC)

{{Outdent|5}} I changed the reference link of the infinite-dimension page, so it should be available for download. Thanks to [[User:Michael D. Turnbull|Mike Turnbull]] advice.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 13:41, 14 December 2020 (UTC)

I was able to find out the weak holomorphic.

Weak definition <ref group=Ifaptmbrttp>Lawrence A. Harris, ''[https://www.ms.uky.edu/~larry/paper.dir/korea.ps Fixed Point Theorems for Infinite Dimensional Holomorphic Functions]'' (undated).</ref>

:A function <math>h:D\rightarrow Y</math> is holomorphic if it is locally bounded and if for each <math>x\in D</math>, <math>y\in X</math> and linear functional <math>\ell\in Y^{\ast}</math>, the function <math>f(\lambda)=\ell (h(x+\lambda y))</math> is holomorphic at <math>\lambda=0</math>.

Since it says ''[[Infinite-dimensional holomorphy|useful criterion]]'', the holomorphic on this page may mean a weak holomorphic. I've read that the reason why holomorphy has a stronger meaning than real variables is that it has an unlimited approach to holomorphic points compared to real numbers. I may need to add a description of the <math>C^n</math> space to make the space we are Integrate more clear. I try read it again without knowing it. Thanks!--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 13:39, 15 December 2020 (UTC)

=== References ===

{{reflist|group=Ifaptmbrttp}}