# A Note on Completeness of Weighted Normed Spaces of Analytic Functions

A Note on Completeness of Weighted Normed Spaces of Analytic Functions Given a non-negative weight v, not necessarily bounded or strictly positive, defined on a domain G in the complex plane, we consider the weighted space $${{H}_{v}^{\infty }}(G)$$ H v ∞ ( G ) of all holomorphic functions on G such that the product v|f| is bounded in G and study the question of when such a space is complete under the canonical sup-seminorm. We obtain both some necessary and some sufficient conditions in terms of the weight v, exhibit several relevant examples, and characterize completeness in the case of spaces with radial weights on balanced domains. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Results in Mathematics Springer Journals

# A Note on Completeness of Weighted Normed Spaces of Analytic Functions

, Volume 72 (2) – May 27, 2017
17 pages

/lp/springer_journal/a-note-on-completeness-of-weighted-normed-spaces-of-analytic-functions-nCY7IW3UIX
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1422-6383
eISSN
1420-9012
D.O.I.
10.1007/s00025-017-0696-2
Publisher site
See Article on Publisher Site

### Abstract

Given a non-negative weight v, not necessarily bounded or strictly positive, defined on a domain G in the complex plane, we consider the weighted space $${{H}_{v}^{\infty }}(G)$$ H v ∞ ( G ) of all holomorphic functions on G such that the product v|f| is bounded in G and study the question of when such a space is complete under the canonical sup-seminorm. We obtain both some necessary and some sufficient conditions in terms of the weight v, exhibit several relevant examples, and characterize completeness in the case of spaces with radial weights on balanced domains.

### Journal

Results in MathematicsSpringer Journals

Published: May 27, 2017

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