IJPAM: Volume 94, No. 2 (2014)

SOME NEW REMARKS ON $l^{=p}$ AND $l^{>p}$ FAMILIES

Michał Różański$^1$, Damian Słota$^2$
Marcin Szweda$^3$, Roman Wituła$^4$
$^{1,2,3,4}$Institute of Mathematics
Silesian University of Technology
Kaszubska 23, 44-100 Gliwice, POLAND


Abstract. Wituła and Słota have distinguished two new subfamilies $l^{>p}$ and $l^{=p}$ of the classical spaces $l^q, \ q>0$, of absolutely $q$-summable sequences (the first one is called the almost $l^p$, the second one - the exactly $l^p$), for every $p>0$. In presented paper the new properties of families $l^{>p}$ and $l^{=p}$, connected with some infinite subfamilies of $l^{>p}$, are introduced.

We also prove that if for an increasing sequence $\{p_n\} $ of positive integers $\varphi (k)$ denotes the number of $\{p_n\} $ not exceeding $k$, i.e. $\varphi$ denotes the counting function of sequence $\{p_n\} $, and $\varphi (k)$ possesses the Chebyshev estimation (like for the prime numbers) then $\{p_n\}\in l^{>1}$ and $\{p_{p_n}\}\in l^{=1}$.

Received: May 4, 2014

AMS Subject Classification: 40A05, 46A99

Key Words and Phrases: almost $l^p$, exactly $l^p$, counting function, prime number theorem

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DOI: 10.12732/ijpam.v94i2.11 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 94
Issue: 2
Pages: 241 - 250

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).