METRICS ON THE SEQUENCE SPACE Λ<i>_p</i> (<i>f</i>)

• HONDA Aoi

  Department of Systems Design and Informatics Kyushu Institute of Technology

• OKAZAKI Yoshiaki

  Department of Systems Design and Informatics Kyushu Institute of Technology

• SATO Hiroshi

  Faculty of Mathematics Kyushu University

Bibliographic Information

Other Title

• METRICS ON THE SEQUENCE SPACE ^|^Lambda;p (f)

Abstract

For 1 ≤ p, r <+∞, f (≠ 0) ∈ L_p (R, dx), and g (≠ 0) ∈ L_r (R, dx), the sequence space Λ_p (f) with metric d^f_p(a,b) was introduced in a previous paper and we discussed the inclusion relations between l_p and Λ_p (f), and the linearity of Λ_p (f).The purpose of this paper is to discuss the topological structures of (Λ_p (f), d^f_p).First we show that the space (Λ_p (f), d^f_p) is a complete separable metric group. Next we show that if Λ_p (f) is a linear space, then (Λ_p (f), d^f_p) is a topological linear space. On the other hand, we give a necessary and sufficient condition for the inclusion Λ_p (f) ⊂ Λ_r (g). Furthermore, we show that the inclusions among the sequence spaces Λ_p (f), Λ_r (g) and l_r are continuous. The fact that Λ_p (f) ⊂ Λ_r (g) as sets implies the continuity of the inclusion (Λ_p (f), d^f_p) → (Λ_r (g), d^g_r) is emphasized.

Journal

• Kyushu Journal of Mathematics

  Kyushu Journal of Mathematics 66 (2), 365-374, 2012

  Faculty of Mathematics, Kyushu University

Keywords

• sequence space Λ<i>_p</i> (<i>f</i>)
• completeness
• linearity
• topological linear space
• <i>l_p</i> space

Details

• CRID: 1390001205227750656
• NII Article ID: 130002543080
• DOI: 10.2206/kyushujm.66.365
• ISSN: 18832032
• ISSN: 13406116
• Web Site: https://www.jstage.jst.go.jp/article/kyushujm/66/2/66_365/_pdf
• Text Lang: en
• Data Source:
  • JaLC
  • Crossref
  • CiNii Articles
• Abstract License Flag: Disallowed