Geodesic
A remark on absolutely convergent series in spaces of germs of analytic functions
Vladikavkazskij matematičeskij žurnal, Tome 17 (2015) no. 3, pp. 53-55 
S. N. Melikhov. A remark on absolutely convergent series in spaces of germs of analytic functions. Vladikavkazskij matematičeskij žurnal, Tome 17 (2015) no. 3, pp. 53-55. http://geodesic.mathdoc.fr/item/VMJ_2015_17_3_a5/
@article{VMJ_2015_17_3_a5,
     author = {S. N. Melikhov},
     title = {A remark on absolutely convergent series in spaces of germs of analytic functions},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {53--55},
     year = {2015},
     volume = {17},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2015_17_3_a5/}
}
TY  - JOUR
AU  - S. N. Melikhov
TI  - A remark on absolutely convergent series in spaces of germs of analytic functions
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2015
SP  - 53
EP  - 55
VL  - 17
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VMJ_2015_17_3_a5/
LA  - ru
ID  - VMJ_2015_17_3_a5
ER  - 
%0 Journal Article
%A S. N. Melikhov
%T A remark on absolutely convergent series in spaces of germs of analytic functions
%J Vladikavkazskij matematičeskij žurnal
%D 2015
%P 53-55
%V 17
%N 3
%U http://geodesic.mathdoc.fr/item/VMJ_2015_17_3_a5/
%G ru
%F VMJ_2015_17_3_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that each absolutely convergent series in the space of germs of all analytic functions on a some set $M\subset\mathbb C^N$ endowed with the projective topology converges absolutely in the Fréchet space of analytic functions on an open neighborhood of $M$. In particular, this allows us to remove the assumptions about the growth of exponents of exponential series, posed in some previous statements.

[1] Makarov B. M., “Ob induktivnykh predelakh normirovannykh prostranstv”, Vestnik LGU, 1965, no. 3(13), 50–58 | MR | Zbl

[2] Melikhov S. N., Momm Z., “O svoistve vnutr-prodolzhaemosti predstavlyayuschikh sistem eksponent na vypuklykh lokalno zamknutykh mnozhestvakh”, Vladikavk. mat. zhurn., 10:2 (2008), 36–45 | MR | Zbl

[3] Bonet J., Meise R., Melikhov S. N., “The dual of the space of holomorphic functions on locally closed convex sets”, Publ. Mat., 49 (2005), 487–509 | DOI | MR | Zbl

[4] Martineau A., “Sur la topologie des espaces de fonctions holomorphes”, Math. Annal., 163 (1966), 62–88 | DOI | MR | Zbl