Some notes on the space of p-summable sequences
Keywords:
Completion, dense, isometric, separable, the space of p-summable sequences.Abstract
Recently, Konca et al. (2015a) have revisited the space p of p -summable sequences of real
numbers and have shown that this space is actually contained in a weighted inner product space called 2 v . In another paper, Konca et al. (2015b) have investigated the space p to show that it is also contained in a weighted 2-inner product space. In this work, we show in the details that p is dense in 2 v as a normed space, and as a 2-normed space. Further, we prove that 2 v is separable and conclude that it is isometric to the completion of p .
References
Bella, A. & Costantini, C. (2015).
Sequential separability vs selective
sequential separability. Filomat, 29(1): 121-
Gähler, S. (1964). Lineare
-normierte räume. Mathematische
Nachrichten, 28(1-2): 1-43.
Gunawan, H. (2001). The space of psummable
sequences and its natural nnorm.
Bulletin of the Australian
Mathematical Society, 64(1): 137-147.
Gunawan, H., Setya-Budhi, W. &
Mashadi, S. Gemawati. (2005). On
volumes of n-dimensional parallelepipeds
on p spaces. Univerzitet u Beogradu
Publikacije Elektrotehnickog Fakulteta.
Serija Matematika, 16:48-54.
Konca, S., Idris, M. & Gunawan, H.
(2015a). p-Summable sequence spaces with
inner products. Bitlis Eren University
Journal of Science and Technology, 5(1):
-41.
Konca, S., Idris, M., Gunawan, H. &
Basarir, M. (2015b). p-Summable
sequence spaces with 2-inner products.
Contemporary Analysis and Applied
Mathematics, 3(2): 213-226.
Konca, S., Idris, M. & Gunawan, H.
(2016). A new 2-inner product on the space
of p-summable sequences. Journal of
Egyptian Mathematical Society, 24: 244-
Konca, S., Gunawan, H. & Basarir, M.
(2014). Some remarks on p as an nnormed
space. Mathematical Sciences
Applications E-Notes, 2(2):45-50.
Retrieved online from
http://dergipark.ulakbim.gov.tr/mathenot/ar
ticle/view/5000105312.
Kreyszig, E. (1989). Introductory
functional analysis with applications. Wiley
Classics Library. Wiley: New York. pp. 21-
, 41-45, 59.
Misiak, A. (1989). n-inner product spaces.
Mathematische Nachrichten, 140: 299-319.
Raj, K., & Sharma, S.K. (2013). Some
new sequence spaces. Applications and
Applied Mathematics, 8(2): 596-613.
Raj, K. & Jamwal, S. (2015). On some
generalized statistically convergent
sequence spaces. Kuwait Journal of
Science, 42(3): 86-104.
Roy, P. (1970). Separability of metric
spaces. Transaction of the American
Mathematical Society, 149(1):19-43.
