Some characterizations of inner product spaces based on angle

Year 2019, , 626 - 632, 15.06.2019

S.m.s. Nabavi Sales

https://doi.org/10.15672/hujms.574044

Abstract

A problem in functional analysis that  arises naturally is about finding necessary and sufficient conditions for a normed space to be an inner product space. By answering this question, mathematicians try to understand the inner product and normed spaces features. In this note, we have discussed this issue and we prove some results concerned with it. We introduce a notion of angle between two vectors in a normed space, denoted by $A_\theta(.,.)$ where $\theta\neq{k\pi\over2}$. We also speak about a notion of orthogonality concerning it, we call it $\theta$-orthogonality.

Keywords

inner product, normed space, angle, orthogonality

References

• D. Amir, Characterizations of Inner Product Spaces, Operator theory: Advances and applications,20, Birkhäuser Verlag, Besel, 1986.
• F. Dadipour and M.S. Moslehian, A characterization of inner product spaces Related to the p-angular distance, J. Math. Anal. App. 371 (11), 677–681, 2010.
• M.M. Day, Some characterizations of inner product spaces, Trans. Amer. Math. Soc. 62, 320–337, 1947.
• M.M. Day, Normed Linear spaces, 3th edition, Springer, New York, 1973.
• C.R. Diminnie, E.Z. Andalafte and R.W. Freese, Angles in normed linear spaces and a characterization of real inner product spaces, Math. Nachr. 129, 197–204, 1986.
• C.R. Diminnie, E.Z. Andalafte and R.W. Freese, Angle bisectors in normed linear spaces, Math. Nachr. 131, 167–173, 1987.
• M. Frećhet, Sur la definition axiomatique d’une classe d’espaces vectoriels distanci´es applicables vectoriellement sur l’espace de Hilbert, Ann. of Math. (2), 36 (3), 705-718,1986.
• R.C. James, Inner product in normed linear spaces, Bull. Amer. Math. Soc. 53, 559–566, 1947.
• R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61, 265–292, 1947.
• P. Jordan and J. von Neumann, On inner product in linear metric spaces, Ann. of Math. (2), 36 (3), 719–723, 1935.
• E.R. Lorch, On certain implications which characterize Hilbert space, Ann. of Math. (2), 49 (2), 523–532, 1948.