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Abstract
Given an n-normed space X for n>= 2$, we investigate the completness of Y (as subspace of X) respect to a new norm that corresponds this new inner product on Y. Moreover, we discuss the angle between two subspaces in Y.
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References
- V. Balestro, A.G. Horv`ath, H. Martini, and R. Teixeira, Angles in normed spaces, ` Aequationes
- Math., 91(2) (2017), 201-236.
- H. Batkunde, and H. Gunawan, A revisit to n-normed spaces through its quotient spaces,
- Matematychni Studii 53(2) (2020), 181-191.
- R. Benitez, Carlos Orthogonality in normed linear spaces: a classification of the different
- concepts and some open problems, Revista Mathematica 2 (1989) 53-57.
- S. G¨ahler, Lineare 2-normierte r¨aume, Math. Nachr. 28 (1964), 1-43.
- S. G¨ahler, Untersuchungen ¨uber verallgemeinerte m-metrische R¨aume. I, Math. Nachr. 40
- (1969), 165-189.
- S. G¨ahler, Untersuchungen ¨uber verallgemeinerte m-metrische R¨aume. II, Math. Nachr. 40
- (1969), 229-264.
- J. R. Giles, Classes of semi-inner-product spaces,Trans. Amer. Math. Soc. 129(3) (1967),
- -446.
- H. Gunawan, On n-inner products, n-norms, and the Cauchy-Schwarz inequality, Sci. Math.
- Jpn. 55 (2002), 53-60.
- H. Gunawan and M. Mashadi, On n-normed spaces, Int. J. Math. Math. Sci., 27 (2001),
- -329.
- H. Gunawan, O. Neswan, and W. Setya-Budhi, A formula for angles between two subspaces
- of inner product spaces, Beitr. Algebra Geom., 46(2) (2005), 311-320
- H. Gunawan, O. Neswan, and E. Sukaesih, Fixed point theorems on bounded sets in an
- n-normed space, J. Math. Comput. Sci., 8(2) (2018), 196-215.
- H. Gunawan and O. Neswan, On angles between subspaces of inner product spaces, J. Indo.
- Math. Soc., 11 (2005), 129-135.
- M. Idris , S. Ekariani and H. Gunawan, On the space of p-summable sequences, Mat. Vesnik.,
- (1) (2013), 58-63.
- R.C. James, Orthogonality in normed linear spaces, Duke Math. J. 12, (1945) 291-302.
- S. Konca, M. Idris, and H. Gunawan, p-summable sequence spaces with inner products, Beu
- J. Sci. Techn., 5(1) (2015), 37-41.
- E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, Inc.,
- New York, 1978.
- A.L. Soenjaya, On n-bounded and n-continuous operator in n-normed space, Journal of the
- Indonesian Mathematical Society, 18(1) (2012), 45-56.
- P. M. Mili`ci`c, On the B-angle and g-angle in normed spaces, J. Inequal. Pure Appl. Math.,
- (3) (2007), 1-9.
- A. Misiak, n-inner product spaces, Math. Nachr. 140 (1989), 299-319.
- M. Nur, M. Idris and Firman, Angle in the space of p-summable sequences, AIMS Mathematics, 7(2) (2022), 2810-2819.
- M. Nur, and H. Gunawan, A new orthogonality and angle in a normed space, Aequationes
- Math. 93 (2019), 547-555.
- M. Nur, and H. Gunawan, A note on the g-angle between subspaces of a normed space,
- Aequationes Math., 95 (2021), 309-318
References
V. Balestro, A.G. Horv`ath, H. Martini, and R. Teixeira, Angles in normed spaces, ` Aequationes
Math., 91(2) (2017), 201-236.
H. Batkunde, and H. Gunawan, A revisit to n-normed spaces through its quotient spaces,
Matematychni Studii 53(2) (2020), 181-191.
R. Benitez, Carlos Orthogonality in normed linear spaces: a classification of the different
concepts and some open problems, Revista Mathematica 2 (1989) 53-57.
S. G¨ahler, Lineare 2-normierte r¨aume, Math. Nachr. 28 (1964), 1-43.
S. G¨ahler, Untersuchungen ¨uber verallgemeinerte m-metrische R¨aume. I, Math. Nachr. 40
(1969), 165-189.
S. G¨ahler, Untersuchungen ¨uber verallgemeinerte m-metrische R¨aume. II, Math. Nachr. 40
(1969), 229-264.
J. R. Giles, Classes of semi-inner-product spaces,Trans. Amer. Math. Soc. 129(3) (1967),
-446.
H. Gunawan, On n-inner products, n-norms, and the Cauchy-Schwarz inequality, Sci. Math.
Jpn. 55 (2002), 53-60.
H. Gunawan and M. Mashadi, On n-normed spaces, Int. J. Math. Math. Sci., 27 (2001),
-329.
H. Gunawan, O. Neswan, and W. Setya-Budhi, A formula for angles between two subspaces
of inner product spaces, Beitr. Algebra Geom., 46(2) (2005), 311-320
H. Gunawan, O. Neswan, and E. Sukaesih, Fixed point theorems on bounded sets in an
n-normed space, J. Math. Comput. Sci., 8(2) (2018), 196-215.
H. Gunawan and O. Neswan, On angles between subspaces of inner product spaces, J. Indo.
Math. Soc., 11 (2005), 129-135.
M. Idris , S. Ekariani and H. Gunawan, On the space of p-summable sequences, Mat. Vesnik.,
(1) (2013), 58-63.
R.C. James, Orthogonality in normed linear spaces, Duke Math. J. 12, (1945) 291-302.
S. Konca, M. Idris, and H. Gunawan, p-summable sequence spaces with inner products, Beu
J. Sci. Techn., 5(1) (2015), 37-41.
E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, Inc.,
New York, 1978.
A.L. Soenjaya, On n-bounded and n-continuous operator in n-normed space, Journal of the
Indonesian Mathematical Society, 18(1) (2012), 45-56.
P. M. Mili`ci`c, On the B-angle and g-angle in normed spaces, J. Inequal. Pure Appl. Math.,
(3) (2007), 1-9.
A. Misiak, n-inner product spaces, Math. Nachr. 140 (1989), 299-319.
M. Nur, M. Idris and Firman, Angle in the space of p-summable sequences, AIMS Mathematics, 7(2) (2022), 2810-2819.
M. Nur, and H. Gunawan, A new orthogonality and angle in a normed space, Aequationes
Math. 93 (2019), 547-555.
M. Nur, and H. Gunawan, A note on the g-angle between subspaces of a normed space,
Aequationes Math., 95 (2021), 309-318