Main Article Content


Associated to Birkhoff orthogonality, we study Birkhoff angles in a normed space and present some of their basic properties. We also discuss how to decide whether an angle is more acute or more obtuse than another. In addition, given two vectors $x$ and $y$ in a normed space, we study the formula for Birkhoff `cosine' of the angle from $x$ to $y$ from which we can, in principal, compute the angle. Some examples will be presented.


Birkhoff angles Birkhoff orthogonality normed spaces

Article Details

Author Biographies

Muhamad Jamaludin, Institut Teknologi Bandung


Mas Daffa Pratamadirdja, Institut Teknologi Bandung

How to Cite
Gunawan, H., Jamaludin, M., & Pratamadirdja, M. D. (2021). On Birkhoff Angles in Normed Spaces. Journal of the Indonesian Mathematical Society, 27(3), 270–284.


  1. J.~Alonso, H. Martini, and S. Wu, ``On Birkhoff orthogonality and isosceles orthogonality
  2. in normed linear spaces'', emph{Aequat. Math.} {bf 83}, No.~2 (2012), 153--189.
  3. V.~Balestro, A.G.~Horv$acute{rm a}$th, H.~Martini, and R.~Teixeira, ``Angles in normed spaces'',
  4. emph{arXiv}:1607.06938v1 [math.MG]
  5. G.~Birkhoff, ``Orthogonality in linear metric spaces'', emph{Duke Math. J.} {bf 1}, No.~2
  6. (1935), 169--172.
  7. H.~Gunawan, J.~Lindiarni, and O.~Neswan,
  8. ``$P-$, $I-$, $g-$, and $D-$ angles in normed spaces'', emph{ITB J. Sci.} {bf 40}A,
  9. No.~1 (2008), 24--32.
  10. M.~Jamaludin, emph{Sudut Birkhoff pada Ruang Bernorma} (in Indonesian), Final Project
  11. Report, Mathematics Undergraduate Program, Bandung Institute of Technology, 2021.
  12. R.C.~James, ``Orthogonality in normed linear spaces'', emph{Duke Math. J.} {bf 12} (1945),
  13. --302.
  14. R.C.~James, ``Orthogonality and linear functionals in normed linear spaces'',
  15. emph{Trans. Amer. Math. Soc.} {bf 61} (1947), 265--292.
  16. P.~Mili$check{rm c}$i$acute{rm c}$, ``On the B-angle and $g$-angle in normed spaces'', emph{JIPAM}
  17. Vol.~{bf 8}, Iss.~4, art.~99 (2007).
  18. J.R.~Partington, ``Orthogonality in normed spaces'', emph{Bull. Austral. Math. Soc.}
  19. {bf 33} (1986), 449--455.
  20. M.D.~Pratamadirdja, emph{Ortogonalitas dan Karakterisasi Sudut Birkhoff di} $ell^1_N$ (in Indonesian),
  21. Final Project Report, Mathematics Undergraduate Program, Bandung Institute of Technology, 2020.
  22. C.~Zhi-Zhi, L.~Wei, L.~L$ddot{rm u}$-lin, and J.~You-qing, ``Projections, Birkhoff orthogonality and
  23. angles in normed spaces'', emph{Comm. Math. Res.} {bf 27}, No.~4 (2011).