# On The Paranormed Spaces of Regularly Convergent Double Sequences

On The Paranormed Spaces of Regularly Convergent Double Sequences In Hardy (Proc Camb Philos 19:86–95, 1916), Hardy defined the normed spaces $$\mathcal {C}_{r}$$ C r and $$\mathcal {C}_{r0}$$ C r 0 of all regularly convergent and regularly null double sequences and made convergence factor calculations (i.e. some beta-dual calculations). In this paper, we extend these spaces to the paranormed spaces $$\mathcal {C}_{r}(t)$$ C r ( t ) and $$\mathcal {C}_{r0}(t)$$ C r 0 ( t ) . Also, we define the paranormed spaces $$\mathcal {C}_{tr}(t)$$ C t r ( t ) and $$\mathcal {C}_{tr0}(t)$$ C t r 0 ( t ) of all totally regularly convergent and totally regularly null double sequences. We examine some topological properties of these spaces and determine their alpha-, beta- and gamma-duals. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Results in Mathematics Springer Journals

# On The Paranormed Spaces of Regularly Convergent Double Sequences

, Volume 72 (2) – May 23, 2017
14 pages

/lp/springer_journal/on-the-paranormed-spaces-of-regularly-convergent-double-sequences-rwP8Mgpl4B
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1422-6383
eISSN
1420-9012
D.O.I.
10.1007/s00025-017-0693-5
Publisher site
See Article on Publisher Site

### Abstract

In Hardy (Proc Camb Philos 19:86–95, 1916), Hardy defined the normed spaces $$\mathcal {C}_{r}$$ C r and $$\mathcal {C}_{r0}$$ C r 0 of all regularly convergent and regularly null double sequences and made convergence factor calculations (i.e. some beta-dual calculations). In this paper, we extend these spaces to the paranormed spaces $$\mathcal {C}_{r}(t)$$ C r ( t ) and $$\mathcal {C}_{r0}(t)$$ C r 0 ( t ) . Also, we define the paranormed spaces $$\mathcal {C}_{tr}(t)$$ C t r ( t ) and $$\mathcal {C}_{tr0}(t)$$ C t r 0 ( t ) of all totally regularly convergent and totally regularly null double sequences. We examine some topological properties of these spaces and determine their alpha-, beta- and gamma-duals.

### Journal

Results in MathematicsSpringer Journals

Published: May 23, 2017

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