A new notion of orthogonality involving area and length

In this paper, a space called geometric space, involving both the notions of area and length, is introduced in general setting. The interplay, between these two ideas, is studied. As a result, a new notion of orthogonality, called area-length orthogonality or A-L orthogonality, is demonstrated. It is shown that A-L orthogonality coincides with the standard notion of orthogonality for inner product spaces. Finally, it is proved that A-L orthogonality implies Birkhoff orthogonality, but not conversely.

Aplikasi Pemetaan terkait Semi-Inner Produk pada Ruang Bernorma Real

In this article, we give the properties of mappings associated with the upper semi-inner product , lower semi-inner product and Lumer semi-inner product which generate the norm on a real normed space. Furthermore, we establish applications to the Birkhoff orthogonality and characterization of best approximants.

Orthogonality in smooth countably normed spaces

We generalize the concepts of normalized duality mapping, J-orthogonality and Birkhoff orthogonality from normed spaces to smooth countably normed spaces. We give some basic properties of J-orthogonality in smooth countably normed spaces and show a relation between J-orthogonality and metric projection on smooth uniformly convex complete countably normed spaces. Moreover, we define the J-dual cone and J-orthogonal complement on a nonempty subset S of a smooth countably normed space and prove some basic results about the J-dual cone and the J-orthogonal complement of S.

On symmetry of strong Birkhoff orthogonality in $$B(\mathcal {H},\mathcal {K})$$ and $$K(\mathcal {H},\mathcal {K})$$

In this paper, complete characterizations of left (or right) symmetric points for strong Birkhoff orthogonality in $$B(\mathcal {H},\mathcal {K})$$ B ( H , K ) and $$K(\mathcal {H},\mathcal {K})$$ K ( H , K ) are given, where $$\mathcal {H},\mathcal {K}$$ H , K are complex Hilbert spaces and $$B(\mathcal {H},\mathcal {K})$$ B ( H , K ) ($$K(\mathcal {H},\mathcal {K})$$ K ( H , K ) ) is the space of all bounded linear (compact) operators from $$\mathcal {H}$$ H into $$\mathcal {K}$$ K .

Relation of Pythagorean and Isosceles Orthogonality with Best approximations in Normed Linear Space

In an arbitrary normed space, though the norm not necessarily coming from the inner product space, the notion of orthogonality may be introduced in various ways as suggested by the mathematicians like R.C. James, B.D. Roberts, G. Birkhoff and S.O. Carlsson. We aim to explore the application of orthogonality in normed linear spaces in the best approximation. Hence it has already been proved that Birkhoff orthogonality implies best approximation and best approximation implies Birkhoff orthogonality. Additionally, it has been proved that in the case of ε -orthogonality, ε -best approximation implies ε -orthogonality and vice-versa. In this article we established relation between Pythagorean orthogonality and best approximation as well as isosceles orthogonality and ε -best approximation in normed space.

LEFT SYMMETRIC POINTS FOR BIRKHOFF ORTHOGONALITY IN THE PREDUALS OF VON NEUMANN ALGEBRAS

In this paper, we give a complete description of left symmetric points for Birkhoff orthogonality in the preduals of von Neumann algebras. As a consequence, except for $\mathbb{C}$, $\ell _{\infty }^{2}$ and $M_{2}(\mathbb{C})$, there are no von Neumann algebras whose preduals have nonzero left symmetric points for Birkhoff orthogonality.